${{ K} _i} _j$ = extrinsic curvature
${{ \gamma} _i} _j$ = spatial metric
$\gamma$ = spatial metric determinant
${{ \hat{\gamma}} _i} _j$ = grid metric
$\hat{\gamma}$ = grid metric determinant
${{ \bar{\gamma}} _i} _j$ = conformal metric
$\bar{\gamma}$ = conformal metric determinant

useful identity:
${{{{ \Gamma} _i} _j} _k} = {{{\frac{1}{2}}} {{\left({{{{{ \gamma} _i} _j} _{,k}} + {{{{ \gamma} _i} _k} _{,j}}{-{{{{ \gamma} _j} _k} _{,i}}}}\right)}}}$
${{{{ \gamma} _i} _j} _{,k}} = {{{{{ \Gamma} _i} _k} _j} + {{{{ \Gamma} _j} _k} _i}}$
${{{{ \bar{\gamma}} _i} _j} _{,k}} = {{{{{ \bar{\Gamma}} _i} _k} _j} + {{{{ \bar{\Gamma}} _j} _k} _i}}$

${{{{ \Gamma} ^i} _j} _k} = {{\frac{1}{2}} {{{{{ \gamma} ^i} ^m}} {{\left({{-{{{{ \gamma} _j} _k} _{,m}}} + {{{{ \gamma} _m} _k} _{,j}} + {{{{ \gamma} _m} _j} _{,k}}}\right)}}}}$

useful identity:
${{\left( {{{{ \gamma} _l} _i}} {{{{ \gamma} ^i} ^j}}\right)} _{,t}} = {0}$
${{{{{{ \gamma} _l} _i}} {{{{{ \gamma} ^i} ^j} _{,t}}}} + {{{{{ \gamma} ^i} ^j}} {{{{{ \gamma} _l} _i} _{,t}}}}} = {0}$
${{{{{{ \gamma} ^l} ^m}} {{{{ \gamma} _l} _i}} {{{{{ \gamma} ^i} ^j} _{,t}}}} + {{{{{ \gamma} ^l} ^m}} {{{{ \gamma} ^i} ^j}} {{{{{ \gamma} _l} _i} _{,t}}}}} = {0}$
${{{{ \gamma} ^i} ^j} _{,t}} = {-{{{{{ \gamma} ^l} ^i}} {{{{ \gamma} ^m} ^j}} {{{{{ \gamma} _l} _m} _{,t}}}}}$

ADM metric evolution:
${{{{ \gamma} _i} _j} _{,t}} = {{{{{-2}} {{\alpha}} \cdot {{{{ K} _i} _j}}} + {{{ \beta} _i} _{,j}} + {{{ \beta} _j} _{,i}}}{-{{{2}} {{{ \beta} ^k}} {{{{{ \Gamma} _k} _i} _j}}}}}$
${{{{ \gamma} _i} _j} _{,t}} = {{{{{ \beta} ^k}} {{{{{ \gamma} _i} _k} _{,j}}}} + {{{{ \beta} ^k}} {{{{{ \gamma} _j} _k} _{,i}}}} + {{{{{ \beta} ^k} _{,j}}} {{{{ \gamma} _i} _k}}} + {{{{{ \beta} ^k} _{,i}}} {{{{ \gamma} _j} _k}}} + {{{-2}} {{\alpha}} \cdot {{{{ K} _i} _j}}} + {{{-2}} {{{ \beta} ^k}} {{{{{ \Gamma} _k} _i} _j}}}}$
${{{{ \gamma} _i} _j} _{,t}} = {{{{{ \beta} ^k}} {{{{{ \gamma} _i} _k} _{,j}}}} + {{{{ \beta} ^k}} {{{{{ \gamma} _j} _k} _{,i}}}} + {{{{{ \beta} ^k} _{,i}}} {{{{ \gamma} _j} _k}}} + {{{{{ \beta} ^k} _{,j}}} {{{{ \gamma} _i} _k}}}{-{{{2}} {{\alpha}} \cdot {{{{ K} _i} _j}}}}{-{{{2}} {{{ \beta} ^k}} {{{{{ \Gamma} _k} _i} _j}}}}}$
${{{{ \gamma} _i} _j} _{,t}} = {{{{{ \beta} ^k}} {{{{{ \gamma} _i} _k} _{,j}}}} + {{{{ \beta} ^k}} {{{{{ \gamma} _j} _k} _{,i}}}}{-{{{{ \beta} ^k}} {{{{{ \gamma} _k} _j} _{,i}}}}} + {{{{ \beta} ^k}} {{{{{ \gamma} _i} _j} _{,k}}}}{-{{{{ \beta} ^k}} {{{{{ \gamma} _k} _i} _{,j}}}}} + {{{{{ \beta} ^k} _{,j}}} {{{{ \gamma} _i} _k}}} + {{{{{ \beta} ^k} _{,i}}} {{{{ \gamma} _j} _k}}}{-{{{2}} {{\alpha}} \cdot {{{{ K} _i} _j}}}}}$
${{{{ \gamma} _i} _j} _{,t}} = {{{{{ \beta} ^k}} {{{{{ \gamma} _i} _j} _{,k}}}} + {{{{{ \beta} ^k} _{,i}}} {{{{ \gamma} _j} _k}}} + {{{{{ \beta} ^k} _{,j}}} {{{{ \gamma} _i} _k}}}{-{{{2}} {{\alpha}} \cdot {{{{ K} _i} _j}}}}}$

${{{{ K} _i} _j} _{,t}} = {{-{{{ \alpha} _{,i}} _{,j}}} + {{{{{{ \Gamma} ^k} _i} _j}} {{{ \alpha} _{,k}}}} + {{{\alpha}} \cdot {{\left({{{{ R} _i} _j} + {{{K}} {{{{ K} _i} _j}}}{-{{{2}} {{{{ K} _i} _k}} {{{{ \gamma} ^k} ^l}} {{{{ K} _l} _j}}}}}\right)}}} + {{{4}} {{\pi}} \cdot {{\alpha}} \cdot {{\left({{{{{{ \gamma} _i} _j}} {{\left({{S}{-{\rho}}}\right)}}}{-{{{2}} {{{{ S} _i} _j}}}}}\right)}}} + {{{{{{ K} _i} _j} _{,k}}} {{{ \beta} ^k}}} + {{{{{ K} _k} _i}} {{{{ \beta} ^k} _{,j}}}} + {{{{{ K} _k} _j}} {{{{ \beta} ^k} _{,i}}}}}$
using ${{{{ \Gamma} ^k} _i} _j} = {{\frac{1}{2}} {{{{{ \gamma} ^k} ^m}} {{\left({{-{{{{ \gamma} _i} _j} _{,m}}} + {{{{ \gamma} _m} _j} _{,i}} + {{{{ \gamma} _m} _i} _{,j}}}\right)}}}}$
${{{{ K} _i} _j} _{,t}} = {{{{-1}} {{{{ \alpha} _{,i}} _{,j}}}} + {{{\alpha}} \cdot {{{{ R} _i} _j}}} + {{{{ \beta} ^k}} {{{{{ K} _i} _j} _{,k}}}} + {{{{{ K} _k} _i}} {{{{ \beta} ^k} _{,j}}}} + {{{{{ K} _k} _j}} {{{{ \beta} ^k} _{,i}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,k}}} {{{{ \gamma} ^k} ^m}} {{{{{ \gamma} _i} _j} _{,m}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,k}}} {{{{ \gamma} ^k} ^m}} {{{{{ \gamma} _m} _j} _{,i}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,k}}} {{{{ \gamma} ^k} ^m}} {{{{{ \gamma} _m} _i} _{,j}}}} + {{{K}} {{\alpha}} \cdot {{{{ K} _i} _j}}} + {{{-8}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _i} _j}}} + {{{-2}} {{\alpha}} \cdot {{{{ K} _i} _k}} {{{{ K} _l} _j}} {{{{ \gamma} ^k} ^l}}} + {{{4}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ \gamma} _i} _j}}} + {{{-4}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}} \cdot {{{{ \gamma} _i} _j}}}}$

Bona-Masso lapse and shift evolution:

lapse evolution:
${{ \alpha} _{,t}} = {{{{{ \alpha} _{,i}}} {{{ \beta} ^i}}}{-{{{{\alpha}^{2}}} {{f}} {{K}}}}}$
using locally-Minkowski normalized coordinates:
${{ \alpha} _{,t}} = {{{{{ \alpha} _{,i}}} {{{ \beta} ^I}} {{{{ e} ^i} _I}}} + {{{-1}} {{K}} {{f}} {{{\alpha}^{2}}}}}$

shift evolution:
${{{ \beta} ^i} _{,t}} = {{ B} ^i}$
using locally-Minkowski normalized coordinates:
${{\left( {{{ \beta} ^I}} {{{{ e} ^i} _I}}\right)} _{,t}} = {{{{ B} ^I}} {{{{ e} ^i} _I}}}$
${{{{{ \beta} ^I}} {{{{{ e} ^i} _I} _{,t}}}} + {{{{{ \beta} ^I} _{,t}}} {{{{ e} ^i} _I}}}} = {{{{ B} ^I}} {{{{ e} ^i} _I}}}$
using ${{{{ e} ^i} _I} _{,t}} = {0}$
${{{{{ \beta} ^I} _{,t}}} {{{{ e} ^i} _I}}} = {{{{ B} ^I}} {{{{ e} ^i} _I}}}$
transforming by inverse of normalization basis
${{{ \beta} ^I} _{,t}} = {{ B} ^I}$

$B^i$ evolution:
${{{ B} ^i} _{,t}} = {{{{\frac{3}{4}}} {{{{ \bar{\Lambda}} ^i} _{,t}}}}{-{{{\eta}} \cdot {{{ B} ^i}}}}}$

using locally-Minkowski normalized coordinates:
${{{{{ B} ^I} _{,t}}} {{{{ e} ^i} _I}}} = {{{{3}} \cdot {{\frac{1}{4}}} {{{{ \bar{\Lambda}} ^I} _{,t}}} {{{{ e} ^i} _I}}} + {{{-1}} {{\eta}} \cdot {{{ B} ^I}} {{{{ e} ^i} _I}}}}$
${{{ B} ^I} _{,t}} = {{{{3}} \cdot {{\frac{1}{4}}} {{{{ \bar{\Lambda}} ^I} _{,t}}}} + {{{-1}} {{\eta}} \cdot {{{ B} ^I}}}}$

conformal $\phi$:
${\phi} = {{{\frac{1}{12}}} {{\log\left( {{\frac{1}{\hat{\gamma}}} {\gamma}}\right)}}}$

conformal $\chi$:
${\chi} = {\sqrt[3]{{\frac{1}{\gamma}} {\hat{\gamma}}}}$

conformal W:
${W} = {{\left({{\frac{1}{\gamma}} {\hat{\gamma}}}\right)}^{\frac{1}{6}}}$
${W} = {\sqrt{\chi}}$
${W} = {\frac{1}{\exp\left({{{2}} {{\phi}}}\right)}}$

conformal metric:
${{{ \bar{\gamma}} _i} _j} = {{{{W}^{2}}} {{{{ \gamma} _i} _j}}}$
${{{ \gamma} _i} _j} = {\frac{{{ \bar{\gamma}} _i} _j}{{W}^{2}}}$

conformal metric inverse:
${{{ \bar{\gamma}} ^i} ^j} = {{{{W}^{-2}}} {{{{ \gamma} ^i} ^j}}}$
${{{ \gamma} ^i} ^j} = {{{{{ \bar{\gamma}} ^i} ^j}} {{{W}^{2}}}}$

conformal metric derivative:
${{{{ \bar{\gamma}} _i} _j} _{,k}} = {{{W}} {{\left({{{{W}} {{{{{ \gamma} _i} _j} _{,k}}}} + {{{2}} {{{ W} _{,k}}} {{{{ \gamma} _i} _j}}}}\right)}}}$
${{{{ \gamma} _i} _j} _{,k}} = {\frac{{{{W}} {{{{{ \bar{\gamma}} _i} _j} _{,k}}}}{-{{{2}} {{{ W} _{,k}}} {{{{ \bar{\gamma}} _i} _j}}}}}{{W}^{3}}}$

conformal metric determinant:
${\bar{\gamma}} = {{{{W}^{6}}} {{\gamma}}}$
${\gamma} = {\frac{\bar{\gamma}}{{W}^{6}}}$

conformal metric constraint:
${\bar{\gamma}} = {\hat{\gamma}}$
${\hat{\gamma}} = {\bar{\gamma}}$

static grid assumption:
${{{{ \hat{\gamma}} _i} _j} _{,t}} = {0}$
${{ \hat{\gamma}} _{,t}} = {0}$

conformal connection:
${{{{ \bar{\Gamma}} _i} _j} _k} = {{{\frac{1}{2}}} {{\left({{{{{ \bar{\gamma}} _i} _j} _{,k}} + {{{{ \bar{\gamma}} _i} _k} _{,j}}{-{{{{ \bar{\gamma}} _j} _k} _{,i}}}}\right)}}}$
${{{{ \bar{\Gamma}} _i} _j} _k} = {{\frac{1}{2}} {{{W}} {{\left({{-{{{W}} {{{{{ \gamma} _j} _k} _{,i}}}}} + {{{W}} {{{{{ \gamma} _i} _k} _{,j}}}} + {{{W}} {{{{{ \gamma} _i} _j} _{,k}}}} + {{{2}} {{{ W} _{,k}}} {{{{ \gamma} _i} _j}}}{-{{{2}} {{{ W} _{,i}}} {{{{ \gamma} _j} _k}}}} + {{{2}} {{{ W} _{,j}}} {{{{ \gamma} _i} _k}}}}\right)}}}}$
${{{{ \bar{\Gamma}} _i} _j} _k} = {{{W}} {{\left({{{{W}} {{{{{ \Gamma} _i} _j} _k}}} + {{{{ W} _{,k}}} {{{{ \gamma} _i} _j}}}{-{{{{ W} _{,i}}} {{{{ \gamma} _j} _k}}}} + {{{{ W} _{,j}}} {{{{ \gamma} _i} _k}}}}\right)}}}$
${{{{ \Gamma} _i} _j} _k} = {{{{{{{ \bar{\Gamma}} _i} _j} _k}} {{\frac{1}{{W}^{2}}}}} + {{{{ W} _{,i}}} {{{{ \gamma} _j} _k}} {{\frac{1}{W}}}} + {{{-1}} {{{ W} _{,k}}} {{{{ \gamma} _i} _j}} {{\frac{1}{W}}}} + {{{-1}} {{{ W} _{,j}}} {{{{ \gamma} _i} _k}} {{\frac{1}{W}}}}}$
${{{{ \Gamma} _i} _j} _k} = {{{{{{{ \bar{\Gamma}} _i} _j} _k}} {{\frac{1}{{W}^{2}}}}} + {{{-1}} {{{ W} _{,j}}} {{{{ \bar{\gamma}} _i} _k}} {{\frac{1}{{W}^{3}}}}} + {{{-1}} {{{ W} _{,k}}} {{{{ \bar{\gamma}} _i} _j}} {{\frac{1}{{W}^{3}}}}} + {{{{ W} _{,i}}} {{{{ \bar{\gamma}} _j} _k}} {{\frac{1}{{W}^{3}}}}}}$
${{{{ \bar{\Gamma}} ^i} _j} _k} = {{{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^i} ^m}} {{{{{ \bar{\gamma}} _m} _j} _{,k}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^i} ^m}} {{{{{ \bar{\gamma}} _m} _k} _{,j}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^i} ^m}} {{{{{ \bar{\gamma}} _j} _k} _{,m}}}}}$
${{{{ \bar{\Gamma}} ^i} _j} _k} = {{\frac{1}{W}}{\left({{{{W}} {{{{{ \Gamma} ^i} _j} _k}}} + {{{{ W} _{,j}}} {{{{ δ} ^i} _k}}} + {{{{ W} _{,k}}} {{{{ δ} ^i} _j}}}{-{{{{ W} _{,m}}} {{{{ \gamma} ^i} ^m}} {{{{ \gamma} _j} _k}}}}}\right)}}$
${{{{ \Gamma} ^i} _j} _k} = {{\frac{1}{W}}{\left({{{{W}} {{{{{ \bar{\Gamma}} ^i} _j} _k}}}{-{{{{ W} _{,k}}} {{{{ δ} ^i} _j}}}}{-{{{{ W} _{,j}}} {{{{ δ} ^i} _k}}}} + {{{{ W} _{,m}}} {{{{ \gamma} ^i} ^m}} {{{{ \gamma} _j} _k}}}}\right)}}$
${{{{ \Gamma} ^i} _j} _k} = {{\frac{1}{W}}{\left({{{{W}} {{{{{ \bar{\Gamma}} ^i} _j} _k}}}{-{{{{ W} _{,j}}} {{{{ δ} ^i} _k}}}}{-{{{{ W} _{,k}}} {{{{ δ} ^i} _j}}}} + {{{{ W} _{,m}}} {{{{ \bar{\gamma}} ^i} ^m}} {{{{ \bar{\gamma}} _j} _k}}}}\right)}}$

extrinsic curvature trace:
${K} = {{{{{ K} _i} _j}} {{{{ \gamma} ^i} ^j}}}$

trace-free extrinsic curvature:
${{{ A} _i} _j} = {{{{ K} _i} _j}{-{{{\frac{1}{3}}} {{{{ \gamma} _i} _j}} {{K}}}}}$
${{{ K} _i} _j} = {{\frac{1}{3}}{\left({{{{3}} {{{{ A} _i} _j}}} + {{{K}} {{{{ \gamma} _i} _j}}}}\right)}}$
${{{ A} ^i} ^j} = {{{{ K} ^i} ^j}{-{{{\frac{1}{3}}} {{{{ \gamma} ^i} ^j}} {{K}}}}}$
${{{ K} ^i} ^j} = {{\frac{1}{3}}{\left({{{{3}} {{{{ A} ^i} ^j}}} + {{{K}} {{{{ \gamma} ^i} ^j}}}}\right)}}$

conformal trace-free extrinsic curvature:
${{{ \bar{A}} _i} _j} = {{{{W}^{2}}} {{{{ A} _i} _j}}}$
${{{ A} _i} _j} = {{{{{ \bar{A}} _i} _j}} {{\frac{1}{{W}^{2}}}}}$
${{{{{ \bar{A}} _i} _j}} {{{{ \bar{\gamma}} ^j} ^n}} {{{{ \bar{\gamma}} ^m} ^i}}} = {{{{{ A} _i} _j}} {{{{ \bar{\gamma}} ^j} ^n}} {{{{ \bar{\gamma}} ^m} ^i}} {{{W}^{2}}}}$
${{{ \bar{A}} ^m} ^n} = {{{{{ A} _i} _j}} {{{{ \gamma} ^j} ^n}} {{{{ \gamma} ^m} ^i}} {{\frac{1}{{W}^{2}}}}}$
${{{ \bar{A}} ^i} ^j} = {\frac{{{ A} ^i} ^j}{{W}^{2}}}$
${{{ A} ^i} ^j} = {{{{{ \bar{A}} ^i} ^j}} {{{W}^{2}}}}$
${{{ K} _i} _j} = {{{{{{ \bar{A}} _i} _j}} {{\frac{1}{{W}^{2}}}}} + {{{\frac{1}{3}}} {{K}} {{{{ \bar{\gamma}} _i} _j}} {{\frac{1}{{W}^{2}}}}}}$

conformal trace-free extrinsic curvature constraint:
${{{ \bar{A}} ^i} _i} = {0}$

trace-free extrinsic curvature derivative:
${{{{ K} _i} _j} _{,k}} = {{\frac{1}{3}}{\left({{{{3}} {{{{{ A} _i} _j} _{,k}}}} + {{{K}} {{{{{ \gamma} _i} _j} _{,k}}}} + {{{{ K} _{,k}}} {{{{ \gamma} _i} _j}}}}\right)}}$

Jacobi identity:
${{{{ \bar{\Gamma}} ^j} _j} _i} = {{\frac{1}{\bar{\gamma}}} {{{\frac{1}{2}}} {{{ \bar{\gamma}} _{,i}}}}}$
${{{{{ \bar{\gamma}} ^j} ^k}} {{{{{ \bar{\gamma}} _j} _k} _{,i}}}} = {{\frac{1}{\bar{\gamma}}} {{ \bar{\gamma}} _{,i}}}$

conformal W evolution:
${{ W} _{,t}} = {\frac{{{{\gamma}} \cdot {{{ \hat{\gamma}} _{,t}}}}{-{{{\hat{\gamma}}} \cdot {{{ \gamma} _{,t}}}}}}{{{6}} {{{\gamma}^{\frac{7}{6}}}} {{{\hat{\gamma}}^{\frac{5}{6}}}}}}$
assuming ${{ \hat{\gamma}} _{,t}} = {0}$
${{ W} _{,t}} = {-{\frac{{{{ \gamma} _{,t}}} {{{\hat{\gamma}}^{\frac{1}{6}}}}}{{{6}} {{{\gamma}^{\frac{7}{6}}}}}}}$
${{ W} _{,t}} = {-{\frac{{{W}} {{{ \gamma} _{,t}}}}{{{6}} {{\gamma}}}}}$
using ${{ \gamma} _{,t}} = {{{\gamma}} \cdot {{{{ \gamma} ^i} ^j}} {{{{{ \gamma} _i} _j} _{,t}}}}$
${{ W} _{,t}} = {-{{\frac{1}{6}} {{{W}} {{{{ \gamma} ^i} ^j}} {{{{{ \gamma} _i} _j} _{,t}}}}}}$
using ${{{{ \gamma} _i} _j} _{,t}} = {{{{{ \beta} ^k}} {{{{{ \gamma} _i} _j} _{,k}}}} + {{{{{ \beta} ^k} _{,i}}} {{{{ \gamma} _j} _k}}} + {{{{{ \beta} ^k} _{,j}}} {{{{ \gamma} _i} _k}}}{-{{{2}} {{\alpha}} \cdot {{{{ K} _i} _j}}}}}$
${{ W} _{,t}} = {{{{-1}} \cdot {{\frac{1}{6}}} {{W}} {{{{ \gamma} ^i} ^j}} {{{ \beta} ^k}} {{{{{ \gamma} _i} _j} _{,k}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{W}} {{{{ \gamma} ^i} ^j}} {{{{ \beta} ^k} _{,i}}} {{{{ \gamma} _j} _k}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{W}} {{{{ \gamma} ^i} ^j}} {{{{ \beta} ^k} _{,j}}} {{{{ \gamma} _i} _k}}} + {{{2}} \cdot {{\frac{1}{6}}} {{W}} {{{{ \gamma} ^i} ^j}} {{\alpha}} \cdot {{{{ K} _i} _j}}}}$
using ${{{{{ \gamma} ^i} ^j}} {{{{ K} _i} _j}}} = {K}$
${{ W} _{,t}} = {{\frac{1}{6}} {{{W}} {{\left({{{{2}} {{K}} {{\alpha}}}{-{{{{ \beta} ^k}} {{{{ \gamma} ^i} ^j}} {{{{{ \gamma} _i} _j} _{,k}}}}}{-{{{{{ \beta} ^k} _{,i}}} {{{{ \gamma} ^i} ^j}} {{{{ \gamma} _j} _k}}}}{-{{{{{ \beta} ^k} _{,j}}} {{{{ \gamma} ^i} ^j}} {{{{ \gamma} _i} _k}}}}}\right)}}}}$
using ${{{ \gamma} _i} _j} = {\frac{{{ \bar{\gamma}} _i} _j}{{W}^{2}}}$ , ${{{ \gamma} ^i} ^j} = {{{{{ \bar{\gamma}} ^i} ^j}} {{{W}^{2}}}}$ , ${{{{ \gamma} _i} _j} _{,k}} = {\frac{{{{W}} {{{{{ \bar{\gamma}} _i} _j} _{,k}}}}{-{{{2}} {{{ W} _{,k}}} {{{{ \bar{\gamma}} _i} _j}}}}}{{W}^{3}}}$
${{ W} _{,t}} = {{{{\frac{1}{3}}} {{K}} {{W}} {{\alpha}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{W}} {{{ \beta} ^k}} {{{{ \bar{\gamma}} ^i} ^j}} {{{{{ \bar{\gamma}} _i} _j} _{,k}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{W}} {{{{ \bar{\gamma}} ^i} ^j}} {{{{ \bar{\gamma}} _i} _k}} {{{{ \beta} ^k} _{,j}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{W}} {{{{ \bar{\gamma}} ^i} ^j}} {{{{ \bar{\gamma}} _j} _k}} {{{{ \beta} ^k} _{,i}}}} + {{{\frac{1}{3}}} {{{ W} _{,k}}} {{{ \beta} ^k}} {{{{ \bar{\gamma}} ^i} ^j}} {{{{ \bar{\gamma}} _i} _j}}}}$
simplifying
${{ W} _{,t}} = {{{{-1}} \cdot {{\frac{1}{3}}} {{W}} {{{{ \beta} ^a} _{,a}}}} + {{{\frac{1}{3}}} {{K}} {{W}} {{\alpha}}} + {{{\frac{1}{3}}} {{{ W} _{,a}}} {{{ \beta} ^a}} {{{{ δ} ^b} _b}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{W}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{\gamma}} _b} _c} _{,a}}}}}$
using ${{{ δ} ^b} _b} = {3}$
${{ W} _{,t}} = {{\frac{1}{6}}{\left({{-{{{2}} {{W}} {{{{ \beta} ^i} _{,i}}}}} + {{{6}} {{{ W} _{,i}}} {{{ \beta} ^i}}} + {{{2}} {{K}} {{W}} {{\alpha}}}{-{{{W}} {{{ \beta} ^i}} {{{{ \bar{\gamma}} ^j} ^k}} {{{{{ \bar{\gamma}} _j} _k} _{,i}}}}}}\right)}}$
using ${{{{{ \bar{\gamma}} ^j} ^k}} {{{{{ \bar{\gamma}} _j} _k} _{,i}}}} = {{\frac{1}{\bar{\gamma}}} {{ \bar{\gamma}} _{,i}}}$
${{ W} _{,t}} = {{{{-1}} \cdot {{\frac{1}{6}}} {{W}} {{{ \bar{\gamma}} _{,i}}} {{{ \beta} ^i}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{W}} {{{{ \beta} ^i} _{,i}}}} + {{{{ W} _{,i}}} {{{ \beta} ^i}}} + {{{\frac{1}{3}}} {{K}} {{W}} {{\alpha}}}}$

using locally-Minkowski normalized coordinates:
${{ W} _{,t}} = {{{{-1}} \cdot {{\frac{1}{6}}} {{W}} {{{ \bar{\gamma}} _{,i}}} {{{ \beta} ^I}} {{{{ e} ^i} _I}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{3}}} {{K}} {{W}} {{\alpha}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{W}} {{{ \beta} ^I}} {{{{{ e} ^i} _I} _{,i}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{W}} {{{{ \beta} ^I} _{,i}}} {{{{ e} ^i} _I}}} + {{{{ W} _{,i}}} {{{ \beta} ^I}} {{{{ e} ^i} _I}}}}$

conformal metric evolution:
${\left( {{{ \bar{\gamma}} _i} _j} = {{{{W}^{2}}} {{{{ \gamma} _i} _j}}}\right)} _{,t}$
distributing derivative...
${{{{ \bar{\gamma}} _i} _j} _{,t}} = {{{W}} {{\left({{{{W}} {{{{{ \gamma} _i} _j} _{,t}}}} + {{{2}} {{{ W} _{,t}}} {{{{ \gamma} _i} _j}}}}\right)}}}$
using ${{{{ \gamma} _i} _j} _{,t}} = {{{{{ \beta} ^k}} {{{{{ \gamma} _i} _j} _{,k}}}} + {{{{{ \beta} ^k} _{,i}}} {{{{ \gamma} _j} _k}}} + {{{{{ \beta} ^k} _{,j}}} {{{{ \gamma} _i} _k}}}{-{{{2}} {{\alpha}} \cdot {{{{ K} _i} _j}}}}}$
${{{{ \bar{\gamma}} _i} _j} _{,t}} = {{{W}} {{\left({{{{W}} {{{ \beta} ^k}} {{{{{ \gamma} _i} _j} _{,k}}}} + {{{W}} {{{{ \beta} ^k} _{,i}}} {{{{ \gamma} _j} _k}}} + {{{W}} {{{{ \beta} ^k} _{,j}}} {{{{ \gamma} _i} _k}}} + {{{2}} {{{ W} _{,t}}} {{{{ \gamma} _i} _j}}}{-{{{2}} {{W}} {{\alpha}} \cdot {{{{ K} _i} _j}}}}}\right)}}}$
using ${{ W} _{,t}} = {{{{-1}} \cdot {{\frac{1}{6}}} {{W}} {{{ \bar{\gamma}} _{,i}}} {{{ \beta} ^i}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{W}} {{{{ \beta} ^i} _{,i}}}} + {{{{ W} _{,i}}} {{{ \beta} ^i}}} + {{{\frac{1}{3}}} {{K}} {{W}} {{\alpha}}}}$
${{{{ \bar{\gamma}} _i} _j} _{,t}} = {{{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,k}}} {{{ \beta} ^k}} {{{{ \gamma} _i} _j}} {{{W}^{2}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \beta} ^k} _{,k}}} {{{{ \gamma} _i} _j}} {{{W}^{2}}}} + {{{{ \beta} ^k}} {{{{{ \gamma} _i} _j} _{,k}}} {{{W}^{2}}}} + {{{{{ \beta} ^k} _{,i}}} {{{{ \gamma} _j} _k}} {{{W}^{2}}}} + {{{{{ \beta} ^k} _{,j}}} {{{{ \gamma} _i} _k}} {{{W}^{2}}}} + {{{-2}} {{\alpha}} \cdot {{{{ K} _i} _j}} {{{W}^{2}}}} + {{{2}} {{W}} {{{ W} _{,k}}} {{{ \beta} ^k}} {{{{ \gamma} _i} _j}}} + {{{2}} \cdot {{\frac{1}{3}}} {{K}} {{\alpha}} \cdot {{{{ \gamma} _i} _j}} {{{W}^{2}}}}}$
using ${{{ \gamma} _i} _j} = {\frac{{{ \bar{\gamma}} _i} _j}{{W}^{2}}}$
${{{{ \bar{\gamma}} _i} _j} _{,t}} = {{{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,k}}} {{{ \beta} ^k}} {{{{ \bar{\gamma}} _i} _j}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} _i} _j}} {{{{ \beta} ^k} _{,k}}}} + {{{{ \beta} ^k}} {{{{{ \bar{\gamma}} _i} _j} _{,k}}}} + {{{{{ \bar{\gamma}} _i} _k}} {{{{ \beta} ^k} _{,j}}}} + {{{{{ \bar{\gamma}} _j} _k}} {{{{ \beta} ^k} _{,i}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{K}} {{\alpha}} \cdot {{{{ \bar{\gamma}} _i} _j}}} + {{{-2}} {{\alpha}} \cdot {{{{ K} _i} _j}} {{{W}^{2}}}}}$
using ${{{ K} _i} _j} = {{{{{{ \bar{A}} _i} _j}} {{\frac{1}{{W}^{2}}}}} + {{{\frac{1}{3}}} {{K}} {{{{ \bar{\gamma}} _i} _j}} {{\frac{1}{{W}^{2}}}}}}$
${{{{ \bar{\gamma}} _i} _j} _{,t}} = {{{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,k}}} {{{ \beta} ^k}} {{{{ \bar{\gamma}} _i} _j}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} _i} _j}} {{{{ \beta} ^k} _{,k}}}} + {{{{ \beta} ^k}} {{{{{ \bar{\gamma}} _i} _j} _{,k}}}} + {{{{{ \bar{\gamma}} _j} _k}} {{{{ \beta} ^k} _{,i}}}} + {{{{{ \bar{\gamma}} _i} _k}} {{{{ \beta} ^k} _{,j}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _i} _j}}}}$

conformal metric perturbation:
${{{ \bar{\epsilon}} _i} _j} = {{{{ \bar{\gamma}} _i} _j}{-{{{ \hat{\gamma}} _i} _j}}}$

conformal metric perturbation spatial derivative:
${{{{ \bar{\epsilon}} _i} _j} _{,k}} = {{{{{ \bar{\gamma}} _i} _j} _{,k}}{-{{{{ \hat{\gamma}} _i} _j} _{,k}}}}$

conformal metric perturbation evolution:
${{{{ \bar{\epsilon}} _i} _j} _{,t}} = {{{{{ \bar{\gamma}} _i} _j} _{,t}}{-{{{{ \hat{\gamma}} _i} _j} _{,t}}}}$
assuming ${{{{ \hat{\gamma}} _i} _j} _{,t}} = {0}$
${{{{ \bar{\epsilon}} _i} _j} _{,t}} = {{{{ \bar{\gamma}} _i} _j} _{,t}}$

using locally-Minkowski normalized coordinates:
${{{{{ e} _i} ^I}} {{{{ e} _j} ^J}} {{{{{ \bar{\epsilon}} _I} _J} _{,t}}}} = {{{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,k}}} {{{ \beta} ^K}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}} {{{{ e} ^k} _K}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _I} _J}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{ \beta} ^K}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}} {{{{{ e} ^k} _K} _{,k}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} _I} _J}} {{{{ \beta} ^K} _{,k}}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}} {{{{ e} ^k} _K}}} + {{{{ \beta} ^A}} {{{{ \bar{\gamma}} _J} _K}} {{{{ e} _j} ^J}} {{{{ e} _k} ^K}} {{{{{ e} ^k} _A} _{,i}}}} + {{{{ \beta} ^A}} {{{{ \bar{\gamma}} _I} _K}} {{{{ e} _i} ^I}} {{{{ e} _k} ^K}} {{{{{ e} ^k} _A} _{,j}}}} + {{{{ \beta} ^K}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} _i} ^I}} {{{{ e} ^k} _K}} {{{{{ e} _j} ^J} _{,k}}}} + {{{{ \beta} ^K}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} _j} ^J}} {{{{ e} ^k} _K}} {{{{{ e} _i} ^I} _{,k}}}} + {{{{ \beta} ^K}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}} {{{{ e} ^k} _K}} {{{{{ \bar{\gamma}} _I} _J} _{,k}}}} + {{{{{ \bar{\gamma}} _J} _K}} {{{{ \beta} ^A} _{,i}}} {{{{ e} _j} ^J}} {{{{ e} ^k} _A}} {{{{ e} _k} ^K}}} + {{{{{ \bar{\gamma}} _I} _K}} {{{{ \beta} ^A} _{,j}}} {{{{ e} _i} ^I}} {{{{ e} ^k} _A}} {{{{ e} _k} ^K}}}}$

${{{{ \bar{\epsilon}} _I} _J} _{,t}} = {{{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,k}}} {{{ \beta} ^K}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^k} _K}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _I} _J}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{ \beta} ^K}} {{{{ \bar{\gamma}} _I} _J}} {{{{{ e} ^k} _K} _{,k}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} _I} _J}} {{{{ \beta} ^K} _{,k}}} {{{{ e} ^k} _K}}} + {{{{ \beta} ^K}} {{{{ e} ^k} _K}} {{{{{ \bar{\gamma}} _I} _J} _{,k}}}} + {{{{{ \bar{\gamma}} _J} _A}} {{{{ \beta} ^A} _{,i}}} {{{{ e} ^i} _I}}} + {{{{{ \bar{\gamma}} _I} _A}} {{{{ \beta} ^A} _{,j}}} {{{{ e} ^j} _J}}} + {{{{ \beta} ^K}} {{{{ \bar{\gamma}} _M} _J}} {{{{ e} ^i} _I}} {{{{ e} ^k} _K}} {{{{{ e} _i} ^M} _{,k}}}} + {{{{ \beta} ^A}} {{{{ \bar{\gamma}} _J} _K}} {{{{ e} ^i} _I}} {{{{ e} _k} ^K}} {{{{{ e} ^k} _A} _{,i}}}} + {{{{ \beta} ^K}} {{{{ \bar{\gamma}} _I} _N}} {{{{ e} ^j} _J}} {{{{ e} ^k} _K}} {{{{{ e} _j} ^N} _{,k}}}} + {{{{ \beta} ^A}} {{{{ \bar{\gamma}} _I} _K}} {{{{ e} ^j} _J}} {{{{ e} _k} ^K}} {{{{{ e} ^k} _A} _{,j}}}}}$

grid vs conformal connection difference:
${{{{ \bar{\Delta}} ^i} _j} _k} = {{{{{ \bar{\Gamma}} ^i} _j} _k}{-{{{{ \hat{\Gamma}} ^i} _j} _k}}}$
${{ \bar{\Delta}} ^i} = {{{{{{ \bar{\Delta}} ^i} _j} _k}} {{{{ \bar{\gamma}} ^j} ^k}}}$
${ \mathcal{C}} ^i$ = arbitrary constant
${{ \bar{\Lambda}} ^i} = {{{ \bar{\Delta}} ^i} + {{ \mathcal{C}} ^i}}$
${{ \bar{\Delta}} ^i} = {{{ \bar{\Lambda}} ^i}{-{{ \mathcal{C}} ^i}}}$

grid vs conformal connection difference evolution:
assuming ${{{ \mathcal{C}} ^i} _{,t}} = {0}$
${{{ \bar{\Lambda}} ^i} _{,t}} = {{{ \bar{\Delta}} ^i} _{,t}}$
using ${{ \bar{\Delta}} ^i} = {{{{{{ \bar{\Delta}} ^i} _j} _k}} {{{{ \bar{\gamma}} ^j} ^k}}}$
${{{ \bar{\Lambda}} ^i} _{,t}} = {{{{{{ \bar{\gamma}} ^j} ^k}} {{{{{{ \bar{\Delta}} ^i} _j} _k} _{,t}}}} + {{{{{{ \bar{\Delta}} ^i} _j} _k}} {{{{{ \bar{\gamma}} ^j} ^k} _{,t}}}}}$
using ${{{{ \bar{\Delta}} ^i} _j} _k} = {{{{{ \bar{\Gamma}} ^i} _j} _k}{-{{{{ \hat{\Gamma}} ^i} _j} _k}}}$
${{{ \bar{\Lambda}} ^i} _{,t}} = {{{{{{ \bar{\gamma}} ^j} ^k}} {{{{{{ \bar{\Gamma}} ^i} _j} _k} _{,t}}}}{-{{{{{ \bar{\gamma}} ^j} ^k}} {{{{{{ \hat{\Gamma}} ^i} _j} _k} _{,t}}}}} + {{{{{{ \bar{\Gamma}} ^i} _j} _k}} {{{{{ \bar{\gamma}} ^j} ^k} _{,t}}}}{-{{{{{{ \bar{\gamma}} ^j} ^k} _{,t}}} {{{{{ \hat{\Gamma}} ^i} _j} _k}}}}}$
assuming ${{{{{ \hat{\Gamma}} ^i} _j} _k} _{,t}} = {0}$
${{{ \bar{\Lambda}} ^i} _{,t}} = {{{{{{ \bar{\gamma}} ^j} ^k}} {{{{{{ \bar{\Gamma}} ^i} _j} _k} _{,t}}}} + {{{{{{ \bar{\Gamma}} ^i} _j} _k}} {{{{{ \bar{\gamma}} ^j} ^k} _{,t}}}}{-{{{{{{ \bar{\gamma}} ^j} ^k} _{,t}}} {{{{{ \hat{\Gamma}} ^i} _j} _k}}}}}$
using ${{{{ \bar{\Gamma}} ^i} _j} _k} = {{{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^i} ^m}} {{{{{ \bar{\gamma}} _m} _j} _{,k}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^i} ^m}} {{{{{ \bar{\gamma}} _m} _k} _{,j}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^i} ^m}} {{{{{ \bar{\gamma}} _j} _k} _{,m}}}}}$
${{{ \bar{\Lambda}} ^i} _{,t}} = {{{{-1}} {{{{{ \bar{\gamma}} ^j} ^k} _{,t}}} {{{{{ \hat{\Gamma}} ^i} _j} _k}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^i} ^a}} {{{{ \bar{\gamma}} ^j} ^k}} {{{{{{ \bar{\gamma}} _j} _k} _{,a}} _{,t}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^j} ^k}} {{{{{ \bar{\gamma}} _a} _k} _{,j}}} {{{{{ \bar{\gamma}} ^i} ^a} _{,t}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^i} ^a}} {{{{ \bar{\gamma}} ^j} ^k}} {{{{{{ \bar{\gamma}} _a} _j} _{,k}} _{,t}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^i} ^a}} {{{{ \bar{\gamma}} ^j} ^k}} {{{{{{ \bar{\gamma}} _a} _k} _{,j}} _{,t}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^j} ^k}} {{{{{ \bar{\gamma}} ^i} ^a} _{,t}}} {{{{{ \bar{\gamma}} _j} _k} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^i} ^a}} {{{{{ \bar{\gamma}} _j} _k} _{,a}}} {{{{{ \bar{\gamma}} ^j} ^k} _{,t}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^j} ^k}} {{{{{ \bar{\gamma}} _a} _j} _{,k}}} {{{{{ \bar{\gamma}} ^i} ^a} _{,t}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^i} ^a}} {{{{{ \bar{\gamma}} _a} _j} _{,k}}} {{{{{ \bar{\gamma}} ^j} ^k} _{,t}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^i} ^a}} {{{{{ \bar{\gamma}} _a} _k} _{,j}}} {{{{{ \bar{\gamma}} ^j} ^k} _{,t}}}}}$
symmetrize ${{ \bar{\gamma}} _i} _j$
${{{ \bar{\Lambda}} ^i} _{,t}} = {{{{-1}} {{{{{ \bar{\gamma}} ^a} ^b} _{,t}}} {{{{{ \hat{\Gamma}} ^i} _a} _b}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{{ \bar{\gamma}} _b} _c} _{,a}} _{,t}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}} {{{{{ \bar{\gamma}} ^b} ^c} _{,t}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{{ \bar{\gamma}} _a} _c} _{,b}}} {{{{{ \bar{\gamma}} ^c} ^i} _{,t}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}} {{{{{ \bar{\gamma}} ^b} ^c} _{,t}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{{ \bar{\gamma}} _a} _c} _{,b}}} {{{{{ \bar{\gamma}} ^c} ^i} _{,t}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{{ \bar{\gamma}} _a} _b} _{,c}} _{,t}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{{ \bar{\gamma}} _a} _b} _{,c}} _{,t}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{{ \bar{\gamma}} ^c} ^i} _{,t}}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{{ \bar{\gamma}} _b} _c} _{,a}}} {{{{{ \bar{\gamma}} ^b} ^c} _{,t}}}}}$
using ${{{{ \bar{\gamma}} ^i} ^j} _{,t}} = {-{{{{{ \bar{\gamma}} ^l} ^i}} {{{{ \bar{\gamma}} ^m} ^j}} {{{{{ \bar{\gamma}} _l} _m} _{,t}}}}}$
${{{ \bar{\Lambda}} ^i} _{,t}} = {{{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{{ \bar{\gamma}} _b} _c} _{,a}} _{,t}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{{ \bar{\gamma}} _a} _b} _{,c}} _{,t}}}} + {{{{{ \bar{\gamma}} ^d} ^a}} {{{{ \bar{\gamma}} ^e} ^b}} {{{{{ \bar{\gamma}} _d} _e} _{,t}}} {{{{{ \hat{\Gamma}} ^i} _a} _b}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^d} ^c}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}} {{{{{ \bar{\gamma}} _d} _e} _{,t}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^d} ^b}} {{{{ \bar{\gamma}} ^e} ^c}} {{{{{ \bar{\gamma}} _b} _c} _{,a}}} {{{{{ \bar{\gamma}} _d} _e} _{,t}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^d} ^b}} {{{{ \bar{\gamma}} ^e} ^c}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}} {{{{{ \bar{\gamma}} _d} _e} _{,t}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^d} ^c}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{{ \bar{\gamma}} _a} _c} _{,b}}} {{{{{ \bar{\gamma}} _d} _e} _{,t}}}}}$
rearrange time derivative of ${{ \bar{\gamma}} _i} _j$ to come first
${{{ \bar{\Lambda}} ^i} _{,t}} = {{{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{ {{{ \bar{\gamma}} _b} _c} _{,t}} _{,a}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{ {{{ \bar{\gamma}} _a} _b} _{,t}} _{,c}}}} + {{{{{ \bar{\gamma}} ^d} ^a}} {{{{ \bar{\gamma}} ^e} ^b}} {{{{{ \bar{\gamma}} _d} _e} _{,t}}} {{{{{ \hat{\Gamma}} ^i} _a} _b}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^d} ^c}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}} {{{{{ \bar{\gamma}} _d} _e} _{,t}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^d} ^b}} {{{{ \bar{\gamma}} ^e} ^c}} {{{{{ \bar{\gamma}} _b} _c} _{,a}}} {{{{{ \bar{\gamma}} _d} _e} _{,t}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^d} ^b}} {{{{ \bar{\gamma}} ^e} ^c}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}} {{{{{ \bar{\gamma}} _d} _e} _{,t}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^d} ^c}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{{ \bar{\gamma}} _a} _c} _{,b}}} {{{{{ \bar{\gamma}} _d} _e} _{,t}}}}}$
substitute ${{{{ \bar{\gamma}} _i} _j} _{,t}} = {{{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,k}}} {{{ \beta} ^k}} {{{{ \bar{\gamma}} _i} _j}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} _i} _j}} {{{{ \beta} ^k} _{,k}}}} + {{{{ \beta} ^k}} {{{{{ \bar{\gamma}} _i} _j} _{,k}}}} + {{{{{ \bar{\gamma}} _j} _k}} {{{{ \beta} ^k} _{,i}}}} + {{{{{ \bar{\gamma}} _i} _k}} {{{{ \beta} ^k} _{,j}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _i} _j}}}}$
${{{ \bar{\Lambda}} ^i} _{,t}} = {{{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{\left( {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,f}}} {{{ \beta} ^f}} {{{{ \bar{\gamma}} _b} _c}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} _b} _c}} {{{{ \beta} ^f} _{,f}}}} + {{{{ \beta} ^f}} {{{{{ \bar{\gamma}} _b} _c} _{,f}}}} + {{{{{ \bar{\gamma}} _c} _f}} {{{{ \beta} ^f} _{,b}}}} + {{{{{ \bar{\gamma}} _b} _f}} {{{{ \beta} ^f} _{,c}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _b} _c}}}\right)} _{,a}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{\left( {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,f}}} {{{ \beta} ^f}} {{{{ \bar{\gamma}} _a} _b}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} _a} _b}} {{{{ \beta} ^f} _{,f}}}} + {{{{ \beta} ^f}} {{{{{ \bar{\gamma}} _a} _b} _{,f}}}} + {{{{{ \bar{\gamma}} _b} _f}} {{{{ \beta} ^f} _{,a}}}} + {{{{{ \bar{\gamma}} _a} _f}} {{{{ \beta} ^f} _{,b}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _a} _b}}}\right)} _{,c}}}} + {{{{{ \bar{\gamma}} ^d} ^a}} {{{{ \bar{\gamma}} ^e} ^b}} {{\left({{{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,f}}} {{{ \beta} ^f}} {{{{ \bar{\gamma}} _d} _e}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} _d} _e}} {{{{ \beta} ^f} _{,f}}}} + {{{{ \beta} ^f}} {{{{{ \bar{\gamma}} _d} _e} _{,f}}}} + {{{{{ \bar{\gamma}} _e} _f}} {{{{ \beta} ^f} _{,d}}}} + {{{{{ \bar{\gamma}} _d} _f}} {{{{ \beta} ^f} _{,e}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _d} _e}}}}\right)}} {{{{{ \hat{\Gamma}} ^i} _a} _b}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^d} ^c}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}} {{\left({{{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,f}}} {{{ \beta} ^f}} {{{{ \bar{\gamma}} _d} _e}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} _d} _e}} {{{{ \beta} ^f} _{,f}}}} + {{{{ \beta} ^f}} {{{{{ \bar{\gamma}} _d} _e} _{,f}}}} + {{{{{ \bar{\gamma}} _e} _f}} {{{{ \beta} ^f} _{,d}}}} + {{{{{ \bar{\gamma}} _d} _f}} {{{{ \beta} ^f} _{,e}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _d} _e}}}}\right)}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^d} ^b}} {{{{ \bar{\gamma}} ^e} ^c}} {{{{{ \bar{\gamma}} _b} _c} _{,a}}} {{\left({{{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,f}}} {{{ \beta} ^f}} {{{{ \bar{\gamma}} _d} _e}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} _d} _e}} {{{{ \beta} ^f} _{,f}}}} + {{{{ \beta} ^f}} {{{{{ \bar{\gamma}} _d} _e} _{,f}}}} + {{{{{ \bar{\gamma}} _e} _f}} {{{{ \beta} ^f} _{,d}}}} + {{{{{ \bar{\gamma}} _d} _f}} {{{{ \beta} ^f} _{,e}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _d} _e}}}}\right)}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^d} ^b}} {{{{ \bar{\gamma}} ^e} ^c}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}} {{\left({{{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,f}}} {{{ \beta} ^f}} {{{{ \bar{\gamma}} _d} _e}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} _d} _e}} {{{{ \beta} ^f} _{,f}}}} + {{{{ \beta} ^f}} {{{{{ \bar{\gamma}} _d} _e} _{,f}}}} + {{{{{ \bar{\gamma}} _e} _f}} {{{{ \beta} ^f} _{,d}}}} + {{{{{ \bar{\gamma}} _d} _f}} {{{{ \beta} ^f} _{,e}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _d} _e}}}}\right)}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^d} ^c}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{{ \bar{\gamma}} _a} _c} _{,b}}} {{\left({{{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,f}}} {{{ \beta} ^f}} {{{{ \bar{\gamma}} _d} _e}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} _d} _e}} {{{{ \beta} ^f} _{,f}}}} + {{{{ \beta} ^f}} {{{{{ \bar{\gamma}} _d} _e} _{,f}}}} + {{{{{ \bar{\gamma}} _e} _f}} {{{{ \beta} ^f} _{,d}}}} + {{{{{ \bar{\gamma}} _d} _f}} {{{{ \beta} ^f} _{,e}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _d} _e}}}}\right)}}}}$
simplifying
${{{ \bar{\Lambda}} ^i} _{,t}} = {{{{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} _b} _c}} {{{{{ \beta} ^f} _{,f}} _{,a}}}} + {{{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^f} _{,f}}} {{{{{ \bar{\gamma}} _b} _c} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^f}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{{ \bar{\gamma}} _b} _c} _{,f}} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} _c} _f}} {{{{{ \beta} ^f} _{,b}} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} _b} _f}} {{{{{ \beta} ^f} _{,c}} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^f} _{,b}}} {{{{{ \bar{\gamma}} _c} _f} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^f} _{,c}}} {{{{{ \bar{\gamma}} _b} _f} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^f} _{,a}}} {{{{{ \bar{\gamma}} _b} _c} _{,f}}}} + {{{\frac{1}{6}}} {{{ \bar{\gamma}} _{,f}}} {{{ \beta} ^f}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{\gamma}} _b} _c} _{,a}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{6}}} {{{ \bar{\gamma}} _{,f}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} _b} _c}} {{{{ \beta} ^f} _{,a}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{6}}} {{{ \beta} ^f}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} _b} _c}} {{{{ \bar{\gamma}} _{,f}} _{,a}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \bar{\gamma}} _{,f}}} {{{ \beta} ^f}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^c}} {{\frac{1}{{\bar{\gamma}}^{2}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^f} _{,f}}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} _a} _b}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \beta} ^f} _{,f}} _{,c}}}} + {{{\alpha}} \cdot {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{{ \alpha} _{,a}}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}}} + {{{{ \beta} ^f}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{{ \bar{\gamma}} _a} _b} _{,f}} _{,c}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} _b} _f}} {{{{{ \beta} ^f} _{,a}} _{,c}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^f} _{,a}}} {{{{{ \bar{\gamma}} _b} _f} _{,c}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^f} _{,c}}} {{{{{ \bar{\gamma}} _a} _b} _{,f}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^f} _{,b}}} {{{{{ \bar{\gamma}} _a} _f} _{,c}}}} + {{{{{ \bar{\gamma}} _a} _f}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \beta} ^f} _{,b}} _{,c}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,f}}} {{{ \beta} ^f}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,f}}} {{{{ \bar{\gamma}} _a} _b}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^f} _{,c}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \beta} ^f}} {{{{ \bar{\gamma}} _a} _b}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} _{,f}} _{,c}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,c}}} {{{ \bar{\gamma}} _{,f}}} {{{ \beta} ^f}} {{{{ \bar{\gamma}} _a} _b}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{\frac{1}{{\bar{\gamma}}^{2}}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _a} _b} _{,c}}}} + {{{-2}} {{{ \alpha} _{,c}}} {{{{ \bar{A}} _a} _b}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^d} ^a}} {{{{ \bar{\gamma}} _d} _e}} {{{{ \bar{\gamma}} ^e} ^b}} {{{{ \beta} ^f} _{,f}}} {{{{{ \hat{\Gamma}} ^i} _a} _b}}} + {{{{ \beta} ^f}} {{{{ \bar{\gamma}} ^d} ^a}} {{{{ \bar{\gamma}} ^e} ^b}} {{{{{ \bar{\gamma}} _d} _e} _{,f}}} {{{{{ \hat{\Gamma}} ^i} _a} _b}}} + {{{{{ \bar{\gamma}} ^d} ^a}} {{{{ \bar{\gamma}} _d} _f}} {{{{ \bar{\gamma}} ^e} ^b}} {{{{ \beta} ^f} _{,e}}} {{{{{ \hat{\Gamma}} ^i} _a} _b}}} + {{{{{ \bar{\gamma}} ^d} ^a}} {{{{ \bar{\gamma}} ^e} ^b}} {{{{ \bar{\gamma}} _e} _f}} {{{{ \beta} ^f} _{,d}}} {{{{{ \hat{\Gamma}} ^i} _a} _b}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,f}}} {{{ \beta} ^f}} {{{{ \bar{\gamma}} ^d} ^a}} {{{{ \bar{\gamma}} _d} _e}} {{{{ \bar{\gamma}} ^e} ^b}} {{{{{ \hat{\Gamma}} ^i} _a} _b}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^d} ^b}} {{{{ \bar{\gamma}} _d} _e}} {{{{ \bar{\gamma}} ^e} ^c}} {{{{ \beta} ^f} _{,f}}} {{{{{ \bar{\gamma}} _b} _c} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^d} ^c}} {{{{ \bar{\gamma}} _d} _e}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{ \beta} ^f} _{,f}}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}}} + {{{\frac{1}{2}}} {{{ \beta} ^f}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^d} ^b}} {{{{ \bar{\gamma}} ^e} ^c}} {{{{{ \bar{\gamma}} _b} _c} _{,a}}} {{{{{ \bar{\gamma}} _d} _e} _{,f}}}} + {{{\frac{1}{2}}} {{{ \beta} ^f}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^d} ^c}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}} {{{{{ \bar{\gamma}} _d} _e} _{,f}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^d} ^b}} {{{{ \bar{\gamma}} _d} _f}} {{{{ \bar{\gamma}} ^e} ^c}} {{{{ \beta} ^f} _{,e}}} {{{{{ \bar{\gamma}} _b} _c} _{,a}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^d} ^c}} {{{{ \bar{\gamma}} _e} _f}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{ \beta} ^f} _{,d}}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^d} ^c}} {{{{ \bar{\gamma}} _d} _f}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{ \beta} ^f} _{,e}}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^d} ^b}} {{{{ \bar{\gamma}} ^e} ^c}} {{{{ \bar{\gamma}} _e} _f}} {{{{ \beta} ^f} _{,d}}} {{{{{ \bar{\gamma}} _b} _c} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,f}}} {{{ \beta} ^f}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^d} ^b}} {{{{ \bar{\gamma}} _d} _e}} {{{{ \bar{\gamma}} ^e} ^c}} {{{{{ \bar{\gamma}} _b} _c} _{,a}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,f}}} {{{ \beta} ^f}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^d} ^c}} {{{{ \bar{\gamma}} _d} _e}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _d} _e}} {{{{ \bar{\gamma}} ^d} ^a}} {{{{ \bar{\gamma}} ^e} ^b}} {{{{{ \hat{\Gamma}} ^i} _a} _b}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^d} ^b}} {{{{ \bar{\gamma}} _d} _e}} {{{{ \bar{\gamma}} ^e} ^c}} {{{{ \beta} ^f} _{,f}}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^d} ^c}} {{{{ \bar{\gamma}} _d} _e}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{ \beta} ^f} _{,f}}} {{{{{ \bar{\gamma}} _a} _c} _{,b}}}} + {{{-1}} {{\alpha}} \cdot {{{{ \bar{A}} _d} _e}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^d} ^c}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}}} + {{{-1}} {{\alpha}} \cdot {{{{ \bar{A}} _d} _e}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^d} ^b}} {{{{ \bar{\gamma}} ^e} ^c}} {{{{{ \bar{\gamma}} _b} _c} _{,a}}}} + {{{-1}} {{{ \beta} ^f}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^d} ^c}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{{ \bar{\gamma}} _a} _c} _{,b}}} {{{{{ \bar{\gamma}} _d} _e} _{,f}}}} + {{{-1}} {{{ \beta} ^f}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^d} ^b}} {{{{ \bar{\gamma}} ^e} ^c}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}} {{{{{ \bar{\gamma}} _d} _e} _{,f}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^d} ^b}} {{{{ \bar{\gamma}} _d} _f}} {{{{ \bar{\gamma}} ^e} ^c}} {{{{ \beta} ^f} _{,e}}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^d} ^c}} {{{{ \bar{\gamma}} _e} _f}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{ \beta} ^f} _{,d}}} {{{{{ \bar{\gamma}} _a} _c} _{,b}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^d} ^c}} {{{{ \bar{\gamma}} _d} _f}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{ \beta} ^f} _{,e}}} {{{{{ \bar{\gamma}} _a} _c} _{,b}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^d} ^b}} {{{{ \bar{\gamma}} ^e} ^c}} {{{{ \bar{\gamma}} _e} _f}} {{{{ \beta} ^f} _{,d}}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,f}}} {{{ \beta} ^f}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^d} ^c}} {{{{ \bar{\gamma}} _d} _e}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{{ \bar{\gamma}} _a} _c} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,f}}} {{{ \beta} ^f}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^d} ^b}} {{{{ \bar{\gamma}} _d} _e}} {{{{ \bar{\gamma}} ^e} ^c}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}} {{\frac{1}{\bar{\gamma}}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _d} _e}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^d} ^c}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{{ \bar{\gamma}} _a} _c} _{,b}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _d} _e}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^d} ^b}} {{{{ \bar{\gamma}} ^e} ^c}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}}}}$
tidying indexes
${{{ \bar{\Lambda}} ^i} _{,t}} = {{{{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \beta} ^d} _{,d}} _{,a}}}} + {{{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,d}}} {{{{{ \bar{\gamma}} _b} _c} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{{ \bar{\gamma}} _c} _d} _{,a}} _{,b}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} _b} _d}} {{{{{ \beta} ^d} _{,c}} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} _c} _d}} {{{{{ \beta} ^d} _{,b}} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,a}}} {{{{{ \bar{\gamma}} _b} _c} _{,d}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _b} _d} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,b}}} {{{{{ \bar{\gamma}} _c} _d} _{,a}}}} + {{{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \bar{\gamma}} _c} _d} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} _c} _d}} {{{{ \beta} ^a} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{6}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} _c} _d}} {{{{ \bar{\gamma}} _{,a}} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \bar{\gamma}} _{,b}}} {{{ \beta} ^b}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} _c} _d}} {{\frac{1}{{\bar{\gamma}}^{2}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} _a} _b}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \beta} ^d} _{,d}} _{,c}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,d}}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}}} + {{{\alpha}} \cdot {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{{ \alpha} _{,a}}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{{ \bar{\gamma}} _b} _c} _{,a}} _{,d}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} _b} _d}} {{{{{ \beta} ^d} _{,a}} _{,c}}}} + {{{{{ \bar{\gamma}} _a} _b}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \beta} ^b} _{,c}} _{,d}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,b}}} {{{{{ \bar{\gamma}} _a} _d} _{,c}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,a}}} {{{{{ \bar{\gamma}} _b} _d} _{,c}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _a} _b} _{,d}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \bar{\gamma}} _b} _c} _{,d}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{\gamma}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \beta} ^a} _{,d}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} _{,a}} _{,d}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \bar{\gamma}} _{,b}}} {{{ \beta} ^b}} {{{{ \bar{\gamma}} _c} _d}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{ \bar{\gamma}} ^d} ^a}} {{\frac{1}{{\bar{\gamma}}^{2}}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _a} _b} _{,c}}}} + {{{-2}} {{{ \alpha} _{,a}}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^a}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} _a} _c}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \beta} ^e} _{,e}}} {{{{{ \hat{\Gamma}} ^i} _b} _d}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{{ \bar{\gamma}} _b} _d} _{,a}}} {{{{{ \hat{\Gamma}} ^i} _c} _e}}} + {{{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} _a} _c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \beta} ^c} _{,d}}} {{{{{ \hat{\Gamma}} ^i} _b} _e}}} + {{{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} _c} _e}} {{{{ \beta} ^e} _{,a}}} {{{{{ \hat{\Gamma}} ^i} _b} _d}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} _b} _d}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{{ \hat{\Gamma}} ^i} _c} _e}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} _b} _d}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \beta} ^f} _{,f}}} {{{{{ \bar{\gamma}} _c} _e} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} _c} _e}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{ \beta} ^f} _{,f}}} {{{{{ \bar{\gamma}} _a} _b} _{,d}}}} + {{{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _d} _f} _{,b}}} {{{{{ \bar{\gamma}} _c} _e} _{,a}}}} + {{{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{{ \bar{\gamma}} _b} _c} _{,e}}} {{{{{ \bar{\gamma}} _d} _f} _{,a}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} _e} _f}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{ \beta} ^f} _{,c}}} {{{{{ \bar{\gamma}} _a} _b} _{,d}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} _c} _e}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{ \beta} ^e} _{,f}}} {{{{{ \bar{\gamma}} _a} _b} _{,d}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} _d} _f}} {{{{ \beta} ^f} _{,b}}} {{{{{ \bar{\gamma}} _c} _e} _{,a}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} _b} _d}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{ \beta} ^d} _{,e}}} {{{{{ \bar{\gamma}} _c} _f} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} _c} _e}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _d} _f} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} _d} _f}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{{ \bar{\gamma}} _b} _c} _{,e}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _a} _b}} {{{{ \bar{\gamma}} ^a} ^c}} {{{{ \bar{\gamma}} ^b} ^d}} {{{{{ \hat{\Gamma}} ^i} _c} _d}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} _b} _d}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \beta} ^f} _{,f}}} {{{{{ \bar{\gamma}} _a} _c} _{,e}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} _c} _e}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{ \beta} ^f} _{,f}}} {{{{{ \bar{\gamma}} _a} _d} _{,b}}}} + {{{-1}} {{\alpha}} \cdot {{{{ \bar{A}} _a} _b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} ^a} ^e}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{{ \bar{\gamma}} _c} _d} _{,e}}}} + {{{-1}} {{\alpha}} \cdot {{{{ \bar{A}} _a} _b}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{ \bar{\gamma}} ^a} ^d}} {{{{ \bar{\gamma}} ^b} ^e}} {{{{{ \bar{\gamma}} _d} _e} _{,c}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{{ \bar{\gamma}} _b} _e} _{,c}}} {{{{{ \bar{\gamma}} _d} _f} _{,a}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _b} _d} _{,f}}} {{{{{ \bar{\gamma}} _c} _e} _{,a}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} _b} _d}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{ \beta} ^d} _{,e}}} {{{{{ \bar{\gamma}} _a} _c} _{,f}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} _d} _f}} {{{{ \beta} ^f} _{,b}}} {{{{{ \bar{\gamma}} _a} _c} _{,e}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} _c} _e}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{ \beta} ^e} _{,f}}} {{{{{ \bar{\gamma}} _a} _d} _{,b}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} _e} _f}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{ \beta} ^f} _{,c}}} {{{{{ \bar{\gamma}} _a} _d} _{,b}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} _c} _e}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _b} _d} _{,f}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} _d} _f}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{{ \bar{\gamma}} _b} _e} _{,c}}} {{\frac{1}{\bar{\gamma}}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _a} _b}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{ \bar{\gamma}} ^a} ^d}} {{{{ \bar{\gamma}} ^b} ^e}} {{{{{ \bar{\gamma}} _c} _d} _{,e}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _a} _b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} ^a} ^e}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{{ \bar{\gamma}} _c} _e} _{,d}}}}}$
simplifying
${{{ \bar{\Lambda}} ^i} _{,t}} = {{{{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \beta} ^d} _{,d}} _{,a}}}} + {{{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,d}}} {{{{{ \bar{\gamma}} _b} _c} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{{ \bar{\gamma}} _c} _d} _{,a}} _{,b}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} _c} _d}} {{{{{ \beta} ^d} _{,b}} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} _b} _d}} {{{{{ \beta} ^d} _{,c}} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,b}}} {{{{{ \bar{\gamma}} _c} _d} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _b} _d} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,a}}} {{{{{ \bar{\gamma}} _b} _c} _{,d}}}} + {{{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \bar{\gamma}} _c} _d} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} _c} _d}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \beta} ^a} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{6}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} _c} _d}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \bar{\gamma}} _{,b}}} {{{ \beta} ^b}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} _c} _d}} {{\frac{1}{{\bar{\gamma}}^{2}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,d}}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} _a} _b}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \beta} ^d} _{,d}} _{,c}}}} + {{{\alpha}} \cdot {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{{ \alpha} _{,a}}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{{ \bar{\gamma}} _b} _c} _{,a}} _{,d}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} _b} _d}} {{{{{ \beta} ^d} _{,a}} _{,c}}}} + {{{{{ \bar{\gamma}} _a} _b}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \beta} ^b} _{,c}} _{,d}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,b}}} {{{{{ \bar{\gamma}} _a} _d} _{,c}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _a} _b} _{,d}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,a}}} {{{{{ \bar{\gamma}} _b} _d} _{,c}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \bar{\gamma}} _b} _c} _{,d}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{\gamma}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \beta} ^a} _{,d}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} _{,a}} _{,d}}} {{{{ \bar{\gamma}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \bar{\gamma}} _{,b}}} {{{ \beta} ^b}} {{{{ \bar{\gamma}} _c} _d}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{ \bar{\gamma}} ^d} ^a}} {{\frac{1}{{\bar{\gamma}}^{2}}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _a} _b} _{,c}}}} + {{{-2}} {{{ \alpha} _{,a}}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^a}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} _a} _c}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \beta} ^e} _{,e}}} {{{{{ \hat{\Gamma}} ^i} _b} _d}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{{ \bar{\gamma}} _b} _d} _{,a}}} {{{{{ \hat{\Gamma}} ^i} _c} _e}}} + {{{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} _a} _c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \beta} ^c} _{,d}}} {{{{{ \hat{\Gamma}} ^i} _b} _e}}} + {{{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} _c} _e}} {{{{ \beta} ^e} _{,a}}} {{{{{ \hat{\Gamma}} ^i} _b} _d}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} _b} _d}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{{ \hat{\Gamma}} ^i} _c} _e}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} _b} _d}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \beta} ^f} _{,f}}} {{{{{ \bar{\gamma}} _c} _e} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} _c} _e}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{ \beta} ^f} _{,f}}} {{{{{ \bar{\gamma}} _a} _b} _{,d}}}} + {{{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _c} _e} _{,a}}} {{{{{ \bar{\gamma}} _d} _f} _{,b}}}} + {{{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{{ \bar{\gamma}} _b} _c} _{,e}}} {{{{{ \bar{\gamma}} _d} _f} _{,a}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} _c} _e}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{ \beta} ^e} _{,f}}} {{{{{ \bar{\gamma}} _a} _b} _{,d}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} _d} _f}} {{{{ \beta} ^f} _{,b}}} {{{{{ \bar{\gamma}} _c} _e} _{,a}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} _b} _d}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{ \beta} ^d} _{,e}}} {{{{{ \bar{\gamma}} _c} _f} _{,a}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} _e} _f}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{ \beta} ^f} _{,c}}} {{{{{ \bar{\gamma}} _a} _b} _{,d}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} _c} _e}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _d} _f} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} _d} _f}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{{ \bar{\gamma}} _b} _c} _{,e}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _a} _b}} {{{{ \bar{\gamma}} ^a} ^c}} {{{{ \bar{\gamma}} ^b} ^d}} {{{{{ \hat{\Gamma}} ^i} _c} _d}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} _b} _d}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \beta} ^f} _{,f}}} {{{{{ \bar{\gamma}} _a} _c} _{,e}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} _c} _e}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{ \beta} ^f} _{,f}}} {{{{{ \bar{\gamma}} _a} _d} _{,b}}}} + {{{-1}} {{\alpha}} \cdot {{{{ \bar{A}} _a} _b}} {{{{ \bar{\gamma}} ^a} ^e}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \bar{\gamma}} _c} _d} _{,e}}}} + {{{-1}} {{\alpha}} \cdot {{{{ \bar{A}} _a} _b}} {{{{ \bar{\gamma}} ^a} ^d}} {{{{ \bar{\gamma}} ^b} ^e}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{{ \bar{\gamma}} _d} _e} _{,c}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{{ \bar{\gamma}} _b} _e} _{,c}}} {{{{{ \bar{\gamma}} _d} _f} _{,a}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _b} _d} _{,f}}} {{{{{ \bar{\gamma}} _c} _e} _{,a}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} _e} _f}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{ \beta} ^f} _{,c}}} {{{{{ \bar{\gamma}} _a} _d} _{,b}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} _b} _d}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{ \beta} ^d} _{,e}}} {{{{{ \bar{\gamma}} _a} _c} _{,f}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} _d} _f}} {{{{ \beta} ^f} _{,b}}} {{{{{ \bar{\gamma}} _a} _c} _{,e}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} _c} _e}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{ \beta} ^e} _{,f}}} {{{{{ \bar{\gamma}} _a} _d} _{,b}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} _d} _f}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{{ \bar{\gamma}} _b} _e} _{,c}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} _c} _e}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _b} _d} _{,f}}} {{\frac{1}{\bar{\gamma}}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _a} _b}} {{{{ \bar{\gamma}} ^a} ^e}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \bar{\gamma}} _c} _e} _{,d}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _a} _b}} {{{{ \bar{\gamma}} ^a} ^d}} {{{{ \bar{\gamma}} ^b} ^e}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{{ \bar{\gamma}} _c} _d} _{,e}}}}}$
simplifying bar metrics
${{{ \bar{\Lambda}} ^i} _{,t}} = {{{{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ δ} _c} ^c}} {{{{{ \beta} ^d} _{,d}} _{,a}}}} + {{{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,d}}} {{{{{ \bar{\gamma}} _b} _c} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{{ \bar{\gamma}} _c} _d} _{,a}} _{,b}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{{ \beta} ^d} _{,d}} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{{ \beta} ^d} _{,d}} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,a}}} {{{{{ \bar{\gamma}} _b} _c} _{,d}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _b} _d} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,b}}} {{{{{ \bar{\gamma}} _c} _d} _{,a}}}} + {{{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \bar{\gamma}} _c} _d} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ δ} ^d} _d}} {{{{ \beta} ^a} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{6}}} {{{ \beta} ^i}} {{{{ δ} ^d} _d}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \bar{\gamma}} _{,b}}} {{{ \beta} ^b}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ δ} ^d} _d}} {{\frac{1}{{\bar{\gamma}}^{2}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^i} ^c}} {{{{{ \beta} ^d} _{,d}} _{,c}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,d}}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}}} + {{{\alpha}} \cdot {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{{ \alpha} _{,a}}} {{{{ \bar{A}} ^c} _c}} {{{{ \bar{\gamma}} ^a} ^i}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{{ \bar{\gamma}} _b} _c} _{,a}} _{,d}}}} + {{{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \beta} ^i} _{,c}} _{,d}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{{ \beta} ^d} _{,a}} _{,d}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,b}}} {{{{{ \bar{\gamma}} _a} _d} _{,c}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,a}}} {{{{{ \bar{\gamma}} _b} _d} _{,c}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _a} _b} _{,d}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \bar{\gamma}} _b} _c} _{,d}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{\gamma}} ^i} ^d}} {{{{ \beta} ^a} _{,d}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \beta} ^i}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \bar{\gamma}} _{,b}}} {{{ \beta} ^b}} {{{{ \bar{\gamma}} ^i} ^a}} {{\frac{1}{{\bar{\gamma}}^{2}}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _a} _b} _{,c}}}} + {{{-2}} {{{ \alpha} _{,a}}} {{{{ \bar{A}} ^i} ^a}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \beta} ^e} _{,e}}} {{{{{ \hat{\Gamma}} ^i} _c} _d}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{{ \bar{\gamma}} _b} _d} _{,a}}} {{{{{ \hat{\Gamma}} ^i} _c} _e}}} + {{{{{ \bar{\gamma}} ^d} ^e}} {{{{ \beta} ^c} _{,d}}} {{{{{ \hat{\Gamma}} ^i} _c} _e}}} + {{{{{ \bar{\gamma}} ^a} ^b}} {{{{ \beta} ^e} _{,a}}} {{{{{ \hat{\Gamma}} ^i} _b} _e}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{{ \hat{\Gamma}} ^i} _d} _e}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \beta} ^f} _{,f}}} {{{{{ \bar{\gamma}} _d} _e} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{ \beta} ^f} _{,f}}} {{{{{ \bar{\gamma}} _a} _b} _{,e}}}} + {{{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _c} _e} _{,a}}} {{{{{ \bar{\gamma}} _d} _f} _{,b}}}} + {{{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{{ \bar{\gamma}} _b} _c} _{,e}}} {{{{{ \bar{\gamma}} _d} _f} _{,a}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^f} _{,b}}} {{{{{ \bar{\gamma}} _c} _f} _{,a}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{ \beta} ^d} _{,e}}} {{{{{ \bar{\gamma}} _d} _f} _{,a}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \beta} ^i} _{,c}}} {{{{{ \bar{\gamma}} _a} _b} _{,d}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{ \beta} ^e} _{,f}}} {{{{{ \bar{\gamma}} _a} _b} _{,e}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _e} _f} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{{ \bar{\gamma}} _b} _c} _{,f}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} ^c} ^d}} {{{{{ \hat{\Gamma}} ^i} _c} _d}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \beta} ^f} _{,f}}} {{{{{ \bar{\gamma}} _a} _d} _{,e}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{ \beta} ^f} _{,f}}} {{{{{ \bar{\gamma}} _a} _e} _{,b}}}} + {{{-1}} {{\alpha}} \cdot {{{{ \bar{A}} ^e} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \bar{\gamma}} _c} _d} _{,e}}}} + {{{-1}} {{\alpha}} \cdot {{{{ \bar{A}} ^d} ^e}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{{ \bar{\gamma}} _d} _e} _{,c}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{{ \bar{\gamma}} _b} _e} _{,c}}} {{{{{ \bar{\gamma}} _d} _f} _{,a}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _b} _d} _{,f}}} {{{{{ \bar{\gamma}} _c} _e} _{,a}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \beta} ^i} _{,c}}} {{{{{ \bar{\gamma}} _a} _d} _{,b}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{ \beta} ^e} _{,f}}} {{{{{ \bar{\gamma}} _a} _e} _{,b}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^f} _{,b}}} {{{{{ \bar{\gamma}} _a} _c} _{,f}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{ \beta} ^d} _{,e}}} {{{{{ \bar{\gamma}} _a} _d} _{,f}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _b} _e} _{,f}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{{ \bar{\gamma}} _b} _f} _{,c}}} {{\frac{1}{\bar{\gamma}}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} ^d} ^e}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{{ \bar{\gamma}} _c} _d} _{,e}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} ^e} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \bar{\gamma}} _c} _e} _{,d}}}}}$
simplifying deltas
${{{ \bar{\Lambda}} ^i} _{,t}} = {{{{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{3}} {{{{{ \beta} ^d} _{,d}} _{,a}}}} + {{{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,d}}} {{{{{ \bar{\gamma}} _b} _c} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{{ \bar{\gamma}} _c} _d} _{,a}} _{,b}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{{ \beta} ^d} _{,d}} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{{ \beta} ^d} _{,d}} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,a}}} {{{{{ \bar{\gamma}} _b} _c} _{,d}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _b} _d} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,b}}} {{{{{ \bar{\gamma}} _c} _d} _{,a}}}} + {{{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \bar{\gamma}} _c} _d} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{\gamma}} ^b} ^i}} {{3}} {{{{ \beta} ^a} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{6}}} {{{ \beta} ^i}} {{3}} \cdot {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \bar{\gamma}} _{,b}}} {{{ \beta} ^b}} {{{{ \bar{\gamma}} ^a} ^i}} {{3}} \cdot {{\frac{1}{{\bar{\gamma}}^{2}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^i} ^c}} {{{{{ \beta} ^d} _{,d}} _{,c}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,d}}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}}} + {{{\alpha}} \cdot {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{{ \alpha} _{,a}}} {{{{ \bar{A}} ^c} _c}} {{{{ \bar{\gamma}} ^a} ^i}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{{ \bar{\gamma}} _b} _c} _{,a}} _{,d}}}} + {{{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \beta} ^i} _{,c}} _{,d}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{{ \beta} ^d} _{,a}} _{,d}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,b}}} {{{{{ \bar{\gamma}} _a} _d} _{,c}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,a}}} {{{{{ \bar{\gamma}} _b} _d} _{,c}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _a} _b} _{,d}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \bar{\gamma}} _b} _c} _{,d}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{\gamma}} ^i} ^d}} {{{{ \beta} ^a} _{,d}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \beta} ^i}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \bar{\gamma}} _{,b}}} {{{ \beta} ^b}} {{{{ \bar{\gamma}} ^i} ^a}} {{\frac{1}{{\bar{\gamma}}^{2}}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _a} _b} _{,c}}}} + {{{-2}} {{{ \alpha} _{,a}}} {{{{ \bar{A}} ^i} ^a}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \beta} ^e} _{,e}}} {{{{{ \hat{\Gamma}} ^i} _c} _d}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{{ \bar{\gamma}} _b} _d} _{,a}}} {{{{{ \hat{\Gamma}} ^i} _c} _e}}} + {{{{{ \bar{\gamma}} ^d} ^e}} {{{{ \beta} ^c} _{,d}}} {{{{{ \hat{\Gamma}} ^i} _c} _e}}} + {{{{{ \bar{\gamma}} ^a} ^b}} {{{{ \beta} ^e} _{,a}}} {{{{{ \hat{\Gamma}} ^i} _b} _e}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{{ \hat{\Gamma}} ^i} _d} _e}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \beta} ^f} _{,f}}} {{{{{ \bar{\gamma}} _d} _e} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{ \beta} ^f} _{,f}}} {{{{{ \bar{\gamma}} _a} _b} _{,e}}}} + {{{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _c} _e} _{,a}}} {{{{{ \bar{\gamma}} _d} _f} _{,b}}}} + {{{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{{ \bar{\gamma}} _b} _c} _{,e}}} {{{{{ \bar{\gamma}} _d} _f} _{,a}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^f} _{,b}}} {{{{{ \bar{\gamma}} _c} _f} _{,a}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{ \beta} ^d} _{,e}}} {{{{{ \bar{\gamma}} _d} _f} _{,a}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \beta} ^i} _{,c}}} {{{{{ \bar{\gamma}} _a} _b} _{,d}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{ \beta} ^e} _{,f}}} {{{{{ \bar{\gamma}} _a} _b} _{,e}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _e} _f} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{{ \bar{\gamma}} _b} _c} _{,f}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} ^c} ^d}} {{{{{ \hat{\Gamma}} ^i} _c} _d}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \beta} ^f} _{,f}}} {{{{{ \bar{\gamma}} _a} _d} _{,e}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{ \beta} ^f} _{,f}}} {{{{{ \bar{\gamma}} _a} _e} _{,b}}}} + {{{-1}} {{\alpha}} \cdot {{{{ \bar{A}} ^e} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \bar{\gamma}} _c} _d} _{,e}}}} + {{{-1}} {{\alpha}} \cdot {{{{ \bar{A}} ^d} ^e}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{{ \bar{\gamma}} _d} _e} _{,c}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{{ \bar{\gamma}} _b} _e} _{,c}}} {{{{{ \bar{\gamma}} _d} _f} _{,a}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _b} _d} _{,f}}} {{{{{ \bar{\gamma}} _c} _e} _{,a}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \beta} ^i} _{,c}}} {{{{{ \bar{\gamma}} _a} _d} _{,b}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{ \beta} ^e} _{,f}}} {{{{{ \bar{\gamma}} _a} _e} _{,b}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^f} _{,b}}} {{{{{ \bar{\gamma}} _a} _c} _{,f}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{ \beta} ^d} _{,e}}} {{{{{ \bar{\gamma}} _a} _d} _{,f}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _b} _e} _{,f}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{{ \bar{\gamma}} _b} _f} _{,c}}} {{\frac{1}{\bar{\gamma}}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} ^d} ^e}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{{ \bar{\gamma}} _c} _d} _{,e}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} ^e} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \bar{\gamma}} _c} _e} _{,d}}}}}$
using ${{{ \bar{A}} ^i} _i} = {0}$
${{{ \bar{\Lambda}} ^i} _{,t}} = {{{{\frac{1}{6}}} {{{ \beta} ^i}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^i} ^c}} {{{{{ \beta} ^d} _{,d}} _{,c}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{{ \beta} ^d} _{,a}} _{,d}}}} + {{{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \beta} ^i} _{,c}} _{,d}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{\gamma}} ^i} ^d}} {{{{ \beta} ^a} _{,d}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \bar{\gamma}} _{,b}}} {{{ \beta} ^b}} {{{{ \bar{\gamma}} ^i} ^a}} {{\frac{1}{{\bar{\gamma}}^{2}}}}} + {{{\frac{1}{2}}} {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \beta} ^a} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \bar{\gamma}} _{,a}}} {{{ \bar{\gamma}} _{,b}}} {{{ \beta} ^b}} {{{{ \bar{\gamma}} ^a} ^i}} {{\frac{1}{{\bar{\gamma}}^{2}}}}} + {{{-2}} {{{ \alpha} _{,a}}} {{{{ \bar{A}} ^i} ^a}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \beta} ^e} _{,e}}} {{{{{ \hat{\Gamma}} ^i} _c} _d}}} + {{{{{ \bar{\gamma}} ^d} ^e}} {{{{ \beta} ^c} _{,d}}} {{{{{ \hat{\Gamma}} ^i} _c} _e}}} + {{{{{ \bar{\gamma}} ^a} ^b}} {{{{ \beta} ^e} _{,a}}} {{{{{ \hat{\Gamma}} ^i} _b} _e}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{{ \hat{\Gamma}} ^i} _d} _e}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \beta} ^f} _{,f}}} {{{{{ \bar{\gamma}} _d} _e} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{ \beta} ^f} _{,f}}} {{{{{ \bar{\gamma}} _a} _b} _{,e}}}} + {{{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,d}}} {{{{{ \bar{\gamma}} _b} _c} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{{ \bar{\gamma}} _c} _d} _{,a}} _{,b}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{ \beta} ^d} _{,e}}} {{{{{ \bar{\gamma}} _d} _f} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _b} _d} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,a}}} {{{{{ \bar{\gamma}} _b} _c} _{,d}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,b}}} {{{{{ \bar{\gamma}} _c} _d} _{,a}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \beta} ^i} _{,c}}} {{{{{ \bar{\gamma}} _a} _b} _{,d}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^f} _{,b}}} {{{{{ \bar{\gamma}} _c} _f} _{,a}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{ \beta} ^e} _{,f}}} {{{{{ \bar{\gamma}} _a} _b} _{,e}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _e} _f} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \bar{\gamma}} _c} _d} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{{ \bar{\gamma}} _b} _c} _{,f}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} ^c} ^d}} {{{{{ \hat{\Gamma}} ^i} _c} _d}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \beta} ^f} _{,f}}} {{{{{ \bar{\gamma}} _a} _d} _{,e}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{ \beta} ^f} _{,f}}} {{{{{ \bar{\gamma}} _a} _e} _{,b}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,d}}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}}} + {{{-1}} {{\alpha}} \cdot {{{{ \bar{A}} ^d} ^e}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{{ \bar{\gamma}} _d} _e} _{,c}}}} + {{{-1}} {{\alpha}} \cdot {{{{ \bar{A}} ^e} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \bar{\gamma}} _c} _d} _{,e}}}} + {{{\alpha}} \cdot {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{{ \bar{\gamma}} _b} _c} _{,a}} _{,d}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _a} _b} _{,d}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \beta} ^i} _{,c}}} {{{{{ \bar{\gamma}} _a} _d} _{,b}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{ \beta} ^e} _{,f}}} {{{{{ \bar{\gamma}} _a} _e} _{,b}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^f} _{,b}}} {{{{{ \bar{\gamma}} _a} _c} _{,f}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{ \beta} ^d} _{,e}}} {{{{{ \bar{\gamma}} _a} _d} _{,f}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,b}}} {{{{{ \bar{\gamma}} _a} _d} _{,c}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,a}}} {{{{{ \bar{\gamma}} _b} _d} _{,c}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{{ \bar{\gamma}} _b} _f} _{,c}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _b} _e} _{,f}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \bar{\gamma}} _b} _c} _{,d}}} {{\frac{1}{\bar{\gamma}}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} ^d} ^e}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{{ \bar{\gamma}} _c} _d} _{,e}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} ^e} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \bar{\gamma}} _c} _e} _{,d}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _a} _b} _{,c}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{{ \bar{\gamma}} _b} _d} _{,a}}} {{{{{ \hat{\Gamma}} ^i} _c} _e}}} + {{{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _c} _e} _{,a}}} {{{{{ \bar{\gamma}} _d} _f} _{,b}}}} + {{{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{{ \bar{\gamma}} _b} _c} _{,e}}} {{{{{ \bar{\gamma}} _d} _f} _{,a}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{{ \bar{\gamma}} _b} _e} _{,c}}} {{{{{ \bar{\gamma}} _d} _f} _{,a}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _b} _d} _{,f}}} {{{{{ \bar{\gamma}} _c} _e} _{,a}}}}}$
tidying indexes
${{{ \bar{\Lambda}} ^i} _{,t}} = {{{{\frac{1}{6}}} {{{ \beta} ^i}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^i} ^a}} {{{{{ \beta} ^b} _{,b}} _{,a}}}} + {{{{{ \bar{\gamma}} ^a} ^b}} {{{{{ \beta} ^i} _{,a}} _{,b}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{{ \beta} ^b} _{,a}} _{,b}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{\gamma}} ^i} ^b}} {{{{ \beta} ^a} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \bar{\gamma}} _{,b}}} {{{ \beta} ^b}} {{{{ \bar{\gamma}} ^i} ^a}} {{\frac{1}{{\bar{\gamma}}^{2}}}}} + {{{\frac{1}{2}}} {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \beta} ^a} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \bar{\gamma}} _{,a}}} {{{ \bar{\gamma}} _{,b}}} {{{ \beta} ^b}} {{{{ \bar{\gamma}} ^a} ^i}} {{\frac{1}{{\bar{\gamma}}^{2}}}}} + {{{-2}} {{{ \alpha} _{,a}}} {{{{ \bar{A}} ^i} ^a}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \beta} ^c} _{,c}}} {{{{{ \hat{\Gamma}} ^i} _a} _b}}} + {{{{{ \bar{\gamma}} ^a} ^b}} {{{{ \beta} ^c} _{,a}}} {{{{{ \hat{\Gamma}} ^i} _c} _b}}} + {{{{{ \bar{\gamma}} ^a} ^b}} {{{{ \beta} ^c} _{,a}}} {{{{{ \hat{\Gamma}} ^i} _b} _c}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \hat{\Gamma}} ^i} _b} _c}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{ \beta} ^d} _{,d}}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{{ \bar{\gamma}} _c} _d} _{,a}} _{,b}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,b}}} {{{{{ \bar{\gamma}} _d} _c} _{,a}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \beta} ^i} _{,c}}} {{{{{ \bar{\gamma}} _a} _b} _{,d}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _b} _d} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,a}}} {{{{{ \bar{\gamma}} _b} _c} _{,d}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _a} _b} _{,d}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^i}} {{{{{ \bar{\gamma}} _b} _c} _{,d}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} ^a} ^b}} {{{{{ \hat{\Gamma}} ^i} _a} _b}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{ \beta} ^d} _{,d}}} {{{{{ \bar{\gamma}} _a} _c} _{,b}}}} + {{{-1}} {{\alpha}} \cdot {{{{ \bar{A}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{\gamma}} _b} _c} _{,a}}}} + {{{-1}} {{\alpha}} \cdot {{{{ \bar{A}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}}} + {{{\alpha}} \cdot {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{{ \bar{\gamma}} _b} _c} _{,a}} _{,d}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _a} _b} _{,d}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \beta} ^i} _{,c}}} {{{{{ \bar{\gamma}} _a} _d} _{,b}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,b}}} {{{{{ \bar{\gamma}} _a} _c} _{,d}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _a} _d} _{,b}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,a}}} {{{{{ \bar{\gamma}} _b} _d} _{,c}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^i}} {{{{{ \bar{\gamma}} _b} _d} _{,c}}} {{\frac{1}{\bar{\gamma}}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{{ \bar{\gamma}} _c} _a} _{,b}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{\gamma}} _b} _a} _{,c}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _a} _b} _{,c}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{{ \bar{\gamma}} _b} _d} _{,a}}} {{{{{ \hat{\Gamma}} ^i} _c} _e}}} + {{{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _c} _e} _{,a}}} {{{{{ \bar{\gamma}} _d} _f} _{,b}}}} + {{{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{{ \bar{\gamma}} _b} _c} _{,e}}} {{{{{ \bar{\gamma}} _d} _f} _{,a}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _b} _d} _{,f}}} {{{{{ \bar{\gamma}} _c} _e} _{,a}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{{ \bar{\gamma}} _b} _e} _{,c}}} {{{{{ \bar{\gamma}} _d} _f} _{,a}}}}}$
symmetrize ${{ \bar{\gamma}} _i} _j$
${{{ \bar{\Lambda}} ^i} _{,t}} = {{{{\frac{1}{6}}} {{{ \beta} ^i}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{{ \beta} ^b} _{,a}} _{,b}}}} + {{{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \beta} ^a} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \bar{\gamma}} _{,b}}} {{{ \beta} ^b}} {{{{ \bar{\gamma}} ^a} ^i}} {{\frac{1}{{\bar{\gamma}}^{2}}}}} + {{{{{ \bar{\gamma}} ^a} ^b}} {{{{{ \beta} ^i} _{,a}} _{,b}}}} + {{{-2}} {{{ \alpha} _{,a}}} {{{{ \bar{A}} ^i} ^a}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \beta} ^c} _{,c}}} {{{{{ \hat{\Gamma}} ^i} _a} _b}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \hat{\Gamma}} ^i} _b} _c}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{ \beta} ^d} _{,d}}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{{ \bar{\gamma}} _c} _d} _{,a}} _{,b}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _b} _d} _{,a}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _a} _b} _{,d}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,a}}} {{{{{ \bar{\gamma}} _b} _c} _{,d}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,b}}} {{{{{ \bar{\gamma}} _c} _d} _{,a}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \beta} ^i} _{,c}}} {{{{{ \bar{\gamma}} _a} _b} _{,d}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^i}} {{{{{ \bar{\gamma}} _b} _c} _{,d}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} ^a} ^b}} {{{{{ \hat{\Gamma}} ^i} _a} _b}}} + {{{2}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \beta} ^c} _{,a}}} {{{{{ \hat{\Gamma}} ^i} _b} _c}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{ \beta} ^d} _{,d}}} {{{{{ \bar{\gamma}} _a} _c} _{,b}}}} + {{{-1}} {{\alpha}} \cdot {{{{ \bar{A}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{\gamma}} _b} _c} _{,a}}}} + {{{-1}} {{\alpha}} \cdot {{{{ \bar{A}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}}} + {{{\alpha}} \cdot {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{{ \bar{\gamma}} _b} _c} _{,a}} _{,d}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,b}}} {{{{{ \bar{\gamma}} _a} _c} _{,d}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,a}}} {{{{{ \bar{\gamma}} _b} _d} _{,c}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _a} _b} _{,d}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _a} _d} _{,b}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \beta} ^i} _{,c}}} {{{{{ \bar{\gamma}} _a} _d} _{,b}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^i}} {{{{{ \bar{\gamma}} _b} _d} _{,c}}} {{\frac{1}{\bar{\gamma}}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{{ \bar{\gamma}} _a} _c} _{,b}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _a} _b} _{,c}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{{ \bar{\gamma}} _b} _d} _{,a}}} {{{{{ \hat{\Gamma}} ^i} _c} _e}}} + {{{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{{ \bar{\gamma}} _b} _c} _{,e}}} {{{{{ \bar{\gamma}} _d} _f} _{,a}}}} + {{{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _c} _e} _{,a}}} {{{{{ \bar{\gamma}} _d} _f} _{,b}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{{ \bar{\gamma}} _b} _e} _{,c}}} {{{{{ \bar{\gamma}} _d} _f} _{,a}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _b} _d} _{,f}}} {{{{{ \bar{\gamma}} _c} _e} _{,a}}}}}$
factoring out ${{ \bar{\gamma}} ^i} ^j$ to form ${{ \bar{A}} _i} _j$
${{{ \bar{\Lambda}} ^i} _{,t}} = {{{{\frac{1}{6}}} {{{ \beta} ^i}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{{ \beta} ^b} _{,a}} _{,b}}}} + {{{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \beta} ^a} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \bar{\gamma}} _{,b}}} {{{ \beta} ^b}} {{{{ \bar{\gamma}} ^a} ^i}} {{\frac{1}{{\bar{\gamma}}^{2}}}}} + {{{{{ \bar{\gamma}} ^a} ^b}} {{{{{ \beta} ^i} _{,a}} _{,b}}}} + {{{-2}} {{{ \alpha} _{,a}}} {{{{{{ \bar{A}} _g} _h}} {{{{ \bar{\gamma}} ^i} ^g}} {{{{ \bar{\gamma}} ^a} ^h}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \beta} ^c} _{,c}}} {{{{{ \hat{\Gamma}} ^i} _a} _b}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \hat{\Gamma}} ^i} _b} _c}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{ \beta} ^d} _{,d}}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{{ \bar{\gamma}} _c} _d} _{,a}} _{,b}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _b} _d} _{,a}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _a} _b} _{,d}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,a}}} {{{{{ \bar{\gamma}} _b} _c} _{,d}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,b}}} {{{{{ \bar{\gamma}} _c} _d} _{,a}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \beta} ^i} _{,c}}} {{{{{ \bar{\gamma}} _a} _b} _{,d}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^i}} {{{{{ \bar{\gamma}} _b} _c} _{,d}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} {{\alpha}} \cdot {{{{{{ \bar{A}} _j} _k}} {{{{ \bar{\gamma}} ^a} ^j}} {{{{ \bar{\gamma}} ^b} ^k}}}} {{{{{ \hat{\Gamma}} ^i} _a} _b}}} + {{{2}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \beta} ^c} _{,a}}} {{{{{ \hat{\Gamma}} ^i} _b} _c}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{ \beta} ^d} _{,d}}} {{{{{ \bar{\gamma}} _a} _c} _{,b}}}} + {{{-1}} {{\alpha}} \cdot {{{{{{ \bar{A}} _l} _m}} {{{{ \bar{\gamma}} ^a} ^l}} {{{{ \bar{\gamma}} ^i} ^m}}}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{\gamma}} _b} _c} _{,a}}}} + {{{-1}} {{\alpha}} \cdot {{{{{{ \bar{A}} _n} _o}} {{{{ \bar{\gamma}} ^a} ^n}} {{{{ \bar{\gamma}} ^b} ^o}}}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}}} + {{{\alpha}} \cdot {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{{ \bar{\gamma}} _b} _c} _{,a}} _{,d}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,b}}} {{{{{ \bar{\gamma}} _a} _c} _{,d}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,a}}} {{{{{ \bar{\gamma}} _b} _d} _{,c}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _a} _b} _{,d}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _a} _d} _{,b}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \beta} ^i} _{,c}}} {{{{{ \bar{\gamma}} _a} _d} _{,b}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^i}} {{{{{ \bar{\gamma}} _b} _d} _{,c}}} {{\frac{1}{\bar{\gamma}}}}} + {{{2}} {{\alpha}} \cdot {{{{{{ \bar{A}} _p} _q}} {{{{ \bar{\gamma}} ^a} ^p}} {{{{ \bar{\gamma}} ^i} ^q}}}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}}} + {{{2}} {{\alpha}} \cdot {{{{{{ \bar{A}} _r} _s}} {{{{ \bar{\gamma}} ^a} ^r}} {{{{ \bar{\gamma}} ^b} ^s}}}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{{ \bar{\gamma}} _a} _c} _{,b}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _a} _b} _{,c}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{{ \bar{\gamma}} _b} _d} _{,a}}} {{{{{ \hat{\Gamma}} ^i} _c} _e}}} + {{{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{{ \bar{\gamma}} _b} _c} _{,e}}} {{{{{ \bar{\gamma}} _d} _f} _{,a}}}} + {{{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _c} _e} _{,a}}} {{{{{ \bar{\gamma}} _d} _f} _{,b}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{{ \bar{\gamma}} _b} _e} _{,c}}} {{{{{ \bar{\gamma}} _d} _f} _{,a}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _b} _d} _{,f}}} {{{{{ \bar{\gamma}} _c} _e} _{,a}}}}}$
tidying indexes
${{{ \bar{\Lambda}} ^i} _{,t}} = {{{{\frac{1}{6}}} {{{ \beta} ^i}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{{ \beta} ^b} _{,a}} _{,b}}}} + {{{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \beta} ^a} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \bar{\gamma}} _{,b}}} {{{ \beta} ^b}} {{{{ \bar{\gamma}} ^a} ^i}} {{\frac{1}{{\bar{\gamma}}^{2}}}}} + {{{{{ \bar{\gamma}} ^a} ^b}} {{{{{ \beta} ^i} _{,a}} _{,b}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \beta} ^c} _{,c}}} {{{{{ \hat{\Gamma}} ^i} _a} _b}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \hat{\Gamma}} ^i} _b} _c}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{ \beta} ^d} _{,d}}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{{ \bar{\gamma}} _c} _d} _{,a}} _{,b}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \beta} ^i} _{,c}}} {{{{{ \bar{\gamma}} _a} _b} _{,d}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _a} _b} _{,d}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _b} _d} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,a}}} {{{{{ \bar{\gamma}} _b} _c} _{,d}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,b}}} {{{{{ \bar{\gamma}} _c} _d} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^i}} {{{{{ \bar{\gamma}} _b} _c} _{,d}}} {{\frac{1}{\bar{\gamma}}}}} + {{{2}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \beta} ^c} _{,a}}} {{{{{ \hat{\Gamma}} ^i} _b} _c}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{ \beta} ^d} _{,d}}} {{{{{ \bar{\gamma}} _a} _c} _{,b}}}} + {{{\alpha}} \cdot {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{{ \bar{\gamma}} _b} _c} _{,a}} _{,d}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _a} _d} _{,b}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \beta} ^i} _{,c}}} {{{{{ \bar{\gamma}} _a} _d} _{,b}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _a} _b} _{,d}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,a}}} {{{{{ \bar{\gamma}} _b} _d} _{,c}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,b}}} {{{{{ \bar{\gamma}} _a} _c} _{,d}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^i}} {{{{{ \bar{\gamma}} _b} _d} _{,c}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _a} _b} _{,c}}}} + {{{-2}} {{{ \alpha} _{,a}}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^a} ^c}} {{{{ \bar{\gamma}} ^i} ^b}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{{ \bar{\gamma}} _b} _d} _{,a}}} {{{{{ \hat{\Gamma}} ^i} _c} _e}}} + {{{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{{ \bar{\gamma}} _b} _c} _{,e}}} {{{{{ \bar{\gamma}} _d} _f} _{,a}}}} + {{{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _c} _e} _{,a}}} {{{{{ \bar{\gamma}} _d} _f} _{,b}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _a} _b}} {{{{ \bar{\gamma}} ^c} ^a}} {{{{ \bar{\gamma}} ^d} ^b}} {{{{{ \hat{\Gamma}} ^i} _c} _d}}} + {{{-1}} {{\alpha}} \cdot {{{{ \bar{A}} _a} _b}} {{{{ \bar{\gamma}} ^c} ^a}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} ^i} ^b}} {{{{{ \bar{\gamma}} _d} _e} _{,c}}}} + {{{-1}} {{\alpha}} \cdot {{{{ \bar{A}} _a} _b}} {{{{ \bar{\gamma}} ^c} ^a}} {{{{ \bar{\gamma}} ^d} ^b}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{{ \bar{\gamma}} _c} _d} _{,e}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _b} _d} _{,f}}} {{{{{ \bar{\gamma}} _c} _e} _{,a}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{{ \bar{\gamma}} _b} _e} _{,c}}} {{{{{ \bar{\gamma}} _d} _f} _{,a}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _a} _b}} {{{{ \bar{\gamma}} ^c} ^a}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} ^i} ^b}} {{{{{ \bar{\gamma}} _c} _d} _{,e}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _a} _b}} {{{{ \bar{\gamma}} ^c} ^a}} {{{{ \bar{\gamma}} ^d} ^b}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{{ \bar{\gamma}} _c} _e} _{,d}}}}}$
using ${{{{{ \bar{\gamma}} ^j} ^k}} {{{{{ \bar{\gamma}} _j} _k} _{,i}}}} = {{\frac{1}{\bar{\gamma}}} {{ \bar{\gamma}} _{,i}}}$
${{{ \bar{\Lambda}} ^i} _{,t}} = {{{{\frac{1}{6}}} {{{ \beta} ^i}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{{ \beta} ^b} _{,a}} _{,b}}}} + {{{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \beta} ^a} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \bar{\gamma}} _{,d}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^d} ^i}} {{\frac{1}{{\bar{\gamma}}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \bar{\gamma}} _{,b}}} {{{ \beta} ^b}} {{{{ \bar{\gamma}} ^a} ^i}} {{\frac{1}{{\bar{\gamma}}^{2}}}}} + {{{{{ \bar{\gamma}} ^a} ^b}} {{{{{ \beta} ^i} _{,a}} _{,b}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,c}}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{ \beta} ^d} _{,d}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \bar{\gamma}} _{,d}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \beta} ^d} _{,a}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{2}}} {{{ \bar{\gamma}} _{,d}}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{ \beta} ^d} _{,c}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{2}}} {{{ \bar{\gamma}} _{,d}}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \beta} ^i} _{,c}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \beta} ^c} _{,c}}} {{{{{ \hat{\Gamma}} ^i} _a} _b}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \hat{\Gamma}} ^i} _b} _c}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{{ \bar{\gamma}} _c} _d} _{,a}} _{,b}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,b}}} {{{{{ \bar{\gamma}} _c} _d} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _b} _d} _{,a}}}} + {{{2}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \beta} ^c} _{,a}}} {{{{{ \hat{\Gamma}} ^i} _b} _c}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{ \beta} ^d} _{,d}}} {{{{{ \bar{\gamma}} _a} _c} _{,b}}}} + {{{\alpha}} \cdot {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{{ \bar{\gamma}} _b} _c} _{,a}} _{,d}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _a} _d} _{,b}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,b}}} {{{{{ \bar{\gamma}} _a} _c} _{,d}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,a}}} {{{{{ \bar{\gamma}} _b} _d} _{,c}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \beta} ^i} _{,c}}} {{{{{ \bar{\gamma}} _a} _d} _{,b}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _a} _b} _{,d}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^i}} {{{{{ \bar{\gamma}} _b} _d} _{,c}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{2}}} {{{ \bar{\gamma}} _{,e}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{{ \bar{\gamma}} _d} _f} _{,a}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _a} _b} _{,c}}}} + {{{-2}} {{{ \alpha} _{,a}}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^a} ^c}} {{{{ \bar{\gamma}} ^i} ^b}}} + {{{-1}} {{\alpha}} \cdot {{{ \bar{\gamma}} _{,c}}} {{{{ \bar{A}} _a} _b}} {{{{ \bar{\gamma}} ^c} ^a}} {{{{ \bar{\gamma}} ^i} ^b}} {{\frac{1}{\bar{\gamma}}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{{ \bar{\gamma}} _b} _d} _{,a}}} {{{{{ \hat{\Gamma}} ^i} _c} _e}}} + {{{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _c} _e} _{,a}}} {{{{{ \bar{\gamma}} _d} _f} _{,b}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _a} _b}} {{{{ \bar{\gamma}} ^c} ^a}} {{{{ \bar{\gamma}} ^d} ^b}} {{{{{ \hat{\Gamma}} ^i} _c} _d}}} + {{{-1}} {{\alpha}} \cdot {{{{ \bar{A}} _a} _b}} {{{{ \bar{\gamma}} ^c} ^a}} {{{{ \bar{\gamma}} ^d} ^b}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{{ \bar{\gamma}} _c} _d} _{,e}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{{ \bar{\gamma}} _b} _e} _{,c}}} {{{{{ \bar{\gamma}} _d} _f} _{,a}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _b} _d} _{,f}}} {{{{{ \bar{\gamma}} _c} _e} _{,a}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _a} _b}} {{{{ \bar{\gamma}} ^c} ^a}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} ^i} ^b}} {{{{{ \bar{\gamma}} _c} _d} _{,e}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _a} _b}} {{{{ \bar{\gamma}} ^c} ^a}} {{{{ \bar{\gamma}} ^d} ^b}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{{ \bar{\gamma}} _c} _e} _{,d}}}}}$
tidying indexes
${{{ \bar{\Lambda}} ^i} _{,t}} = {{{{\frac{1}{6}}} {{{ \beta} ^i}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{{ \beta} ^b} _{,a}} _{,b}}}} + {{{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \beta} ^a} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \bar{\gamma}} _{,b}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{\frac{1}{{\bar{\gamma}}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \bar{\gamma}} _{,b}}} {{{ \beta} ^b}} {{{{ \bar{\gamma}} ^a} ^i}} {{\frac{1}{{\bar{\gamma}}^{2}}}}} + {{{{{ \bar{\gamma}} ^a} ^b}} {{{{{ \beta} ^i} _{,a}} _{,b}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \beta} ^b} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{2}}} {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{\gamma}} ^b} ^a}} {{{{ \beta} ^i} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \beta} ^c} _{,c}}} {{{{{ \hat{\Gamma}} ^i} _a} _b}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \hat{\Gamma}} ^i} _b} _c}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{{ \bar{\gamma}} _c} _d} _{,a}} _{,b}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _b} _d} _{,a}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,b}}} {{{{{ \bar{\gamma}} _c} _d} _{,a}}}} + {{{2}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \beta} ^c} _{,a}}} {{{{{ \hat{\Gamma}} ^i} _b} _c}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{ \beta} ^d} _{,d}}} {{{{{ \bar{\gamma}} _a} _c} _{,b}}}} + {{{\alpha}} \cdot {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{{ \bar{\gamma}} _b} _c} _{,a}} _{,d}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,b}}} {{{{{ \bar{\gamma}} _a} _c} _{,d}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,a}}} {{{{{ \bar{\gamma}} _b} _d} _{,c}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \beta} ^i} _{,c}}} {{{{{ \bar{\gamma}} _a} _d} _{,b}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _a} _d} _{,b}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _a} _b} _{,d}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^i}} {{{{{ \bar{\gamma}} _b} _d} _{,c}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{2}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^b}} {{{{ \bar{\gamma}} ^c} ^a}} {{{{ \bar{\gamma}} ^d} ^i}} {{{{{ \bar{\gamma}} _c} _d} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _a} _b} _{,c}}}} + {{{-2}} {{{ \alpha} _{,a}}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^a} ^c}} {{{{ \bar{\gamma}} ^i} ^b}}} + {{{-1}} {{\alpha}} \cdot {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^i} ^c}} {{\frac{1}{\bar{\gamma}}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{{ \bar{\gamma}} _b} _d} _{,a}}} {{{{{ \hat{\Gamma}} ^i} _c} _e}}} + {{{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _c} _e} _{,a}}} {{{{{ \bar{\gamma}} _d} _f} _{,b}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _a} _b}} {{{{ \bar{\gamma}} ^c} ^a}} {{{{ \bar{\gamma}} ^d} ^b}} {{{{{ \hat{\Gamma}} ^i} _c} _d}}} + {{{-1}} {{\alpha}} \cdot {{{{ \bar{A}} _a} _b}} {{{{ \bar{\gamma}} ^c} ^a}} {{{{ \bar{\gamma}} ^d} ^b}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{{ \bar{\gamma}} _c} _d} _{,e}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _b} _d} _{,f}}} {{{{{ \bar{\gamma}} _c} _e} _{,a}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{{ \bar{\gamma}} _b} _e} _{,c}}} {{{{{ \bar{\gamma}} _d} _f} _{,a}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _a} _b}} {{{{ \bar{\gamma}} ^c} ^a}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} ^i} ^b}} {{{{{ \bar{\gamma}} _c} _d} _{,e}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _a} _b}} {{{{ \bar{\gamma}} ^c} ^a}} {{{{ \bar{\gamma}} ^d} ^b}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{{ \bar{\gamma}} _c} _e} _{,d}}}}}$

using locally-Minkowski normalized coordinates:
${{{{{ \bar{\Lambda}} ^I} _{,t}}} {{{{ e} ^i} _I}}} = {{{{\frac{1}{6}}} {{{ \beta} ^I}} {{{{ e} ^i} _I}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{3}}} {{{ \beta} ^B}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^i} _I}} {{{{{{ e} ^b} _B} _{,a}} _{,b}}}} + {{{\frac{1}{3}}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^i} _I}} {{{{{ \beta} ^B} _{,a}} _{,b}}}} + {{{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ e} ^b} _B}} {{{{ e} ^i} _I}} {{{{{ e} ^a} _A} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \beta} ^A} _{,b}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^i} _I}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \bar{\gamma}} _{,b}}} {{{ \beta} ^B}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^i} _I}} {{\frac{1}{{\bar{\gamma}}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \bar{\gamma}} _{,b}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^i} _I}} {{\frac{1}{{\bar{\gamma}}^{2}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{ \beta} ^C}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ e} ^i} _I}} {{{{{ \hat{\Gamma}} ^I} _A} _B}} {{{{{ e} ^c} _C} _{,c}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \beta} ^C} _{,c}}} {{{{ e} ^c} _C}} {{{{ e} ^i} _I}} {{{{{ \hat{\Gamma}} ^I} _A} _B}}} + {{{{ \beta} ^I}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{{{ e} ^i} _I} _{,a}} _{,b}}}} + {{{{{ \bar{\gamma}} ^A} ^B}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^i} _I}} {{{{{ \beta} ^I} _{,a}} _{,b}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^i} _I}} {{{{{ \hat{\Gamma}} ^I} _B} _C}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^B}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^i} _I}} {{{{{ e} ^b} _B} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \beta} ^B} _{,b}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^i} _I}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{2}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^I}} {{{{ \bar{\gamma}} ^B} ^A}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{{ e} ^i} _I} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{2}}} {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{\gamma}} ^B} ^A}} {{{{ \beta} ^I} _{,b}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^i} _I}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _A} _B}} {{{{ \bar{\gamma}} ^C} ^A}} {{{{ \bar{\gamma}} ^D} ^B}} {{{{ e} ^i} _I}} {{{{{ \hat{\Gamma}} ^I} _C} _D}}} + {{{-2}} {{{ \alpha} _{,a}}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^A} ^C}} {{{{ \bar{\gamma}} ^I} ^B}} {{{{ e} ^a} _A}} {{{{ e} ^i} _I}}} + {{{-1}} {{\alpha}} \cdot {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} ^I} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^i} _I}} {{\frac{1}{\bar{\gamma}}}}} + {{{2}} {{{ \beta} ^C}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ e} ^a} _A}} {{{{ e} _c} ^D}} {{{{ e} ^i} _I}} {{{{{ \hat{\Gamma}} ^I} _B} _D}} {{{{{ e} ^c} _C} _{,a}}}} + {{{2}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \beta} ^C} _{,a}}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{ e} _c} ^D}} {{{{ e} ^i} _I}} {{{{{ \hat{\Gamma}} ^I} _B} _D}}} + {{{\alpha}} \cdot {{{{ \bar{A}} _D} _E}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _c} ^E}} {{{{ e} ^i} _I}} {{{{{ e} _b} ^D} _{,a}}}} + {{{\alpha}} \cdot {{{{ \bar{A}} _D} _E}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^D}} {{{{ e} ^c} _C}} {{{{ e} ^i} _I}} {{{{{ e} _c} ^E} _{,a}}}} + {{{\alpha}} \cdot {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^D}} {{{{ e} ^c} _C}} {{{{ e} _c} ^E}} {{{{ e} ^i} _I}} {{{{{ \bar{A}} _D} _E} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} _E} _F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} _d} ^F}} {{{{ e} ^i} _I}} {{{{{{ e} _c} ^E} _{,a}} _{,b}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} _E} _F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _c} ^E}} {{{{ e} ^d} _D}} {{{{ e} ^i} _I}} {{{{{{ e} _d} ^F} _{,a}} _{,b}}}} + {{{\frac{1}{2}}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} _E} _F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _c} ^E}} {{{{ e} ^i} _I}} {{{{{ e} ^d} _D} _{,b}}} {{{{{ e} _d} ^F} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} _E} _F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^E}} {{{{ e} ^c} _C}} {{{{ e} ^i} _I}} {{{{{ e} ^d} _D} _{,c}}} {{{{{ e} _d} ^F} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _c} ^E}} {{{{ e} ^d} _D}} {{{{ e} _d} ^F}} {{{{ e} ^i} _I}} {{{{{{ \bar{\gamma}} _E} _F} _{,a}} _{,b}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} _E} _F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _d} ^F}} {{{{ e} ^i} _I}} {{{{{ e} _b} ^E} _{,a}}} {{{{{ e} ^d} _D} _{,c}}}} + {{{\frac{1}{2}}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} _E} _F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _d} ^F}} {{{{ e} ^i} _I}} {{{{{ e} _c} ^E} _{,a}}} {{{{{ e} ^d} _D} _{,b}}}} + {{{\frac{1}{2}}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _c} ^E}} {{{{ e} _d} ^F}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _E} _F} _{,a}}} {{{{{ e} ^d} _D} _{,b}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^E}} {{{{ e} ^c} _C}} {{{{ e} _d} ^F}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _E} _F} _{,a}}} {{{{{ e} ^d} _D} _{,c}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} _E} _F}} {{{{ \beta} ^D} _{,c}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} _d} ^F}} {{{{ e} ^i} _I}} {{{{{ e} _b} ^E} _{,a}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} _E} _F}} {{{{ \beta} ^D} _{,b}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _c} ^E}} {{{{ e} ^d} _D}} {{{{ e} ^i} _I}} {{{{{ e} _d} ^F} _{,a}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} _E} _F}} {{{{ \beta} ^D} _{,b}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} _d} ^F}} {{{{ e} ^i} _I}} {{{{{ e} _c} ^E} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} _E} _F}} {{{{ \beta} ^D} _{,c}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^E}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} ^i} _I}} {{{{{ e} _d} ^F} _{,a}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \beta} ^D} _{,b}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _c} ^E}} {{{{ e} ^d} _D}} {{{{ e} _d} ^F}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _E} _F} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \beta} ^D} _{,c}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^E}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} _d} ^F}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _E} _F} _{,a}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _D} _E}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} _a} ^D}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} ^i} _I}} {{{{{ e} _b} ^E} _{,c}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _D} _E}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^E}} {{{{ e} ^c} _C}} {{{{ e} ^i} _I}} {{{{{ e} _a} ^D} _{,c}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} _a} ^D}} {{{{ e} ^b} _B}} {{{{ e} _b} ^E}} {{{{ e} ^c} _C}} {{{{ e} ^i} _I}} {{{{{ \bar{A}} _D} _E} _{,c}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} ^C} ^I}} {{{{ \bar{\gamma}} _E} _F}} {{{{ e} ^a} _A}} {{{{ e} _a} ^E}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} ^i} _I}} {{{{{ e} _c} ^F} _{,b}}} {{{{{ e} ^d} _D} _{,d}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} ^C} ^I}} {{{{ \bar{\gamma}} _E} _F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _c} ^F}} {{{{ e} ^i} _I}} {{{{{ e} _a} ^E} _{,b}}} {{{{{ e} ^d} _D} _{,d}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} ^C} ^I}} {{{{ e} ^a} _A}} {{{{ e} _a} ^E}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _c} ^F}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _E} _F} _{,b}}} {{{{{ e} ^d} _D} _{,d}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} ^C} ^I}} {{{{ \bar{\gamma}} _E} _F}} {{{{ \beta} ^D} _{,d}}} {{{{ e} ^a} _A}} {{{{ e} _a} ^E}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} ^i} _I}} {{{{{ e} _c} ^F} _{,b}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} ^C} ^I}} {{{{ \bar{\gamma}} _E} _F}} {{{{ \beta} ^D} _{,d}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _c} ^F}} {{{{ e} ^d} _D}} {{{{ e} ^i} _I}} {{{{{ e} _a} ^E} _{,b}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} ^C} ^I}} {{{{ \beta} ^D} _{,d}}} {{{{ e} ^a} _A}} {{{{ e} _a} ^E}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _c} ^F}} {{{{ e} ^d} _D}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _E} _F} _{,b}}}} + {{{-1}} {{{ \beta} ^I}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} _E} _F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} _d} ^F}} {{{{{ e} _a} ^E} _{,b}}} {{{{{ e} ^i} _I} _{,c}}}} + {{{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^E}} {{{{ e} ^c} _C}} {{{{ e} _d} ^F}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _E} _F} _{,c}}} {{{{{ e} ^d} _D} _{,a}}}} + {{{-1}} {{{ \beta} ^I}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} _E} _F}} {{{{ e} ^a} _A}} {{{{ e} _a} ^E}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{{ e} _d} ^F} _{,b}}} {{{{{ e} ^i} _I} _{,c}}}} + {{{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} ^D} ^E}} {{{{ \bar{\gamma}} _F} _G}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^F}} {{{{ e} ^d} _D}} {{{{ e} ^i} _I}} {{{{{ \hat{\Gamma}} ^I} _C} _E}} {{{{{ e} _d} ^G} _{,a}}}} + {{{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} _a} ^E}} {{{{ e} ^b} _B}} {{{{ e} _b} ^F}} {{{{ e} ^c} _C}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _E} _F} _{,d}}} {{{{{ e} ^d} _D} _{,c}}}} + {{{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} _E} _F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _c} ^F}} {{{{ e} ^d} _D}} {{{{ e} ^i} _I}} {{{{{{ e} _b} ^E} _{,a}} _{,d}}}} + {{{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} _E} _F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^E}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} ^i} _I}} {{{{{{ e} _c} ^F} _{,a}} _{,d}}}} + {{{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^E}} {{{{ e} ^c} _C}} {{{{ e} _c} ^F}} {{{{ e} ^d} _D}} {{{{ e} ^i} _I}} {{{{{{ \bar{\gamma}} _E} _F} _{,a}} _{,d}}}} + {{{-1}} {{{ \beta} ^I}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ e} ^a} _A}} {{{{ e} _a} ^E}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} _d} ^F}} {{{{{ \bar{\gamma}} _E} _F} _{,b}}} {{{{{ e} ^i} _I} _{,c}}}} + {{{-1}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} _a} ^E}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _c} ^F}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _E} _F} _{,d}}} {{{{{ e} ^d} _D} _{,b}}}} + {{{-1}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} ^C} ^I}} {{{{ \bar{\gamma}} _E} _F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _d} ^F}} {{{{ e} ^i} _I}} {{{{{ e} _a} ^E} _{,b}}} {{{{{ e} ^d} _D} _{,c}}}} + {{{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} _E} _F}} {{{{ e} ^a} _A}} {{{{ e} _a} ^E}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} ^i} _I}} {{{{{ e} _b} ^F} _{,d}}} {{{{{ e} ^d} _D} _{,c}}}} + {{{-1}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} ^C} ^I}} {{{{ e} ^a} _A}} {{{{ e} _a} ^E}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _d} ^F}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _E} _F} _{,b}}} {{{{{ e} ^d} _D} _{,c}}}} + {{{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} _E} _F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^E}} {{{{ e} ^c} _C}} {{{{ e} ^i} _I}} {{{{{ e} ^d} _D} _{,a}}} {{{{{ e} _d} ^F} _{,c}}}} + {{{-1}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} ^C} ^I}} {{{{ \bar{\gamma}} _E} _F}} {{{{ e} ^a} _A}} {{{{ e} _a} ^E}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} ^i} _I}} {{{{{ e} ^d} _D} _{,c}}} {{{{{ e} _d} ^F} _{,b}}}} + {{{{ \beta} ^D}} {{{{ \bar{\gamma}} 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{{{{ e} _c} ^F}} {{{{ e} ^d} _D}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _E} _F} _{,d}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} ^D} ^I}} {{{{ \bar{\gamma}} _E} _F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} _d} ^F}} {{{{ e} ^i} _I}} {{{{{ e} _b} ^E} _{,c}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} ^D} ^I}} {{{{ \bar{\gamma}} _E} _F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^E}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} ^i} _I}} {{{{{ e} _d} ^F} _{,c}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} ^D} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^E}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} _d} ^F}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _E} _F} _{,c}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{2}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^B}} {{{{ \bar{\gamma}} ^C} ^A}} {{{{ \bar{\gamma}} ^D} ^I}} {{{{ \bar{\gamma}} _E} _F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} _d} ^F}} {{{{ e} ^i} _I}} {{{{{ e} _c} ^E} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{2}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^B}} {{{{ \bar{\gamma}} ^C} ^A}} {{{{ \bar{\gamma}} ^D} ^I}} {{{{ \bar{\gamma}} _E} _F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _c} ^E}} {{{{ e} ^d} _D}} {{{{ e} ^i} _I}} {{{{{ e} _d} ^F} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{2}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^B}} {{{{ \bar{\gamma}} ^C} ^A}} {{{{ \bar{\gamma}} ^D} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _c} ^E}} {{{{ e} ^d} _D}} {{{{ e} _d} ^F}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _E} _F} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} {{\alpha}} \cdot {{{{ \bar{A}} _A} _B}} {{{{ \bar{\gamma}} ^C} ^A}} {{{{ \bar{\gamma}} ^D} ^B}} {{{{ \bar{\gamma}} ^E} ^I}} {{{{ \bar{\gamma}} _F} _G}} {{{{ e} ^c} _C}} {{{{ e} _c} ^F}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{ e} ^i} _I}} {{{{{ e} _d} ^G} _{,e}}}} + {{{-1}} {{\alpha}} \cdot {{{{ \bar{A}} _A} _B}} {{{{ \bar{\gamma}} ^C} ^A}} {{{{ \bar{\gamma}} ^D} ^B}} {{{{ \bar{\gamma}} ^E} ^I}} {{{{ \bar{\gamma}} _F} _G}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} _d} ^G}} {{{{ e} ^e} _E}} {{{{ e} ^i} _I}} {{{{{ e} _c} ^F} _{,e}}}} + {{{-1}} {{\alpha}} \cdot {{{{ \bar{A}} _A} _B}} {{{{ \bar{\gamma}} ^C} ^A}} {{{{ \bar{\gamma}} ^D} ^B}} {{{{ \bar{\gamma}} ^E} ^I}} {{{{ e} ^c} _C}} {{{{ e} _c} ^F}} {{{{ e} ^d} _D}} {{{{ e} _d} ^G}} {{{{ e} ^e} _E}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _F} _G} _{,e}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _A} _B}} {{{{ \bar{\gamma}} ^C} ^A}} {{{{ \bar{\gamma}} ^D} ^B}} {{{{ \bar{\gamma}} ^E} ^I}} {{{{ \bar{\gamma}} _F} _G}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{ e} _e} ^G}} {{{{ e} ^i} _I}} {{{{{ e} _c} ^F} _{,d}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _A} _B}} {{{{ \bar{\gamma}} ^C} ^A}} {{{{ \bar{\gamma}} ^D} ^E}} {{{{ \bar{\gamma}} _F} _G}} {{{{ \bar{\gamma}} ^I} ^B}} {{{{ e} ^c} _C}} {{{{ e} _c} ^F}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{ e} ^i} _I}} {{{{{ e} _d} ^G} _{,e}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _A} _B}} {{{{ \bar{\gamma}} ^C} ^A}} {{{{ \bar{\gamma}} ^D} ^E}} {{{{ \bar{\gamma}} ^I} ^B}} {{{{ e} ^c} _C}} {{{{ e} _c} ^F}} {{{{ e} ^d} _D}} {{{{ e} _d} ^G}} {{{{ e} ^e} _E}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _F} _G} _{,e}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _A} _B}} {{{{ \bar{\gamma}} ^C} ^A}} {{{{ \bar{\gamma}} ^D} ^B}} {{{{ \bar{\gamma}} ^E} ^I}} {{{{ \bar{\gamma}} _F} _G}} {{{{ e} ^c} _C}} {{{{ e} _c} ^F}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{ e} ^i} _I}} {{{{{ e} _e} ^G} _{,d}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _A} _B}} {{{{ \bar{\gamma}} ^C} ^A}} {{{{ \bar{\gamma}} ^D} ^B}} {{{{ \bar{\gamma}} ^E} ^I}} {{{{ e} ^c} _C}} {{{{ e} _c} ^F}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{ e} _e} ^G}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _F} _G} _{,d}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _A} _B}} {{{{ \bar{\gamma}} ^C} ^A}} {{{{ \bar{\gamma}} ^D} ^E}} {{{{ \bar{\gamma}} _F} _G}} {{{{ \bar{\gamma}} ^I} ^B}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} _d} ^G}} {{{{ e} ^e} _E}} {{{{ e} ^i} _I}} {{{{{ e} _c} ^F} _{,e}}}} + {{{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ \bar{\gamma}} _J} _K}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _c} ^G}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{ e} _e} ^H}} {{{{ e} ^f} _F}} {{{{ e} _f} ^K}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _G} _H} _{,a}}} {{{{{ e} _d} ^J} _{,b}}}} + {{{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ \bar{\gamma}} _G} _H}} {{{{ \bar{\gamma}} _J} _K}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{ e} _e} ^H}} {{{{ e} ^f} _F}} {{{{ e} _f} ^K}} {{{{ e} ^i} _I}} {{{{{ e} _c} ^G} _{,a}}} {{{{{ e} _d} ^J} _{,b}}}} + {{{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ \bar{\gamma}} _G} _H}} {{{{ \bar{\gamma}} _J} _K}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _c} ^G}} {{{{ e} ^d} _D}} {{{{ e} _d} ^J}} {{{{ e} ^e} _E}} {{{{ e} ^f} _F}} {{{{ e} ^i} _I}} {{{{{ e} _e} ^H} _{,a}}} {{{{{ e} _f} ^K} _{,b}}}} + {{{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ \bar{\gamma}} _G} _H}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} _d} ^J}} {{{{ e} ^e} _E}} {{{{ e} _e} ^H}} {{{{ e} ^f} _F}} {{{{ e} _f} ^K}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _J} _K} _{,b}}} {{{{{ e} _c} ^G} _{,a}}}} + {{{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ \bar{\gamma}} _J} _K}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _c} ^G}} {{{{ e} ^d} _D}} {{{{ e} _d} ^J}} {{{{ e} ^e} _E}} {{{{ e} _e} ^H}} {{{{ e} ^f} _F}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _G} _H} _{,a}}} {{{{{ e} _f} ^K} _{,b}}}} + {{{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ \bar{\gamma}} _G} _H}} {{{{ \bar{\gamma}} _J} _K}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _c} ^G}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{ e} ^f} _F}} {{{{ e} _f} ^K}} {{{{ e} ^i} _I}} {{{{{ e} _d} ^J} _{,b}}} {{{{{ e} _e} ^H} _{,a}}}} + {{{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ \bar{\gamma}} _G} _H}} {{{{ \bar{\gamma}} _J} _K}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} _d} ^J}} {{{{ e} ^e} _E}} {{{{ e} _e} ^H}} {{{{ e} ^f} _F}} {{{{ e} ^i} _I}} {{{{{ e} _c} ^G} _{,a}}} {{{{{ e} _f} ^K} _{,b}}}} + {{{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ \bar{\gamma}} _G} _H}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _c} ^G}} {{{{ e} ^d} _D}} {{{{ e} _d} ^J}} {{{{ e} ^e} _E}} {{{{ e} ^f} _F}} {{{{ e} _f} ^K}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _J} _K} _{,b}}} {{{{{ e} _e} ^H} _{,a}}}} + {{{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _c} ^G}} {{{{ e} ^d} _D}} {{{{ e} _d} ^J}} {{{{ e} ^e} _E}} {{{{ e} _e} ^H}} {{{{ e} ^f} _F}} {{{{ e} _f} ^K}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _G} _H} _{,a}}} {{{{{ \bar{\gamma}} _J} _K} _{,b}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ \bar{\gamma}} _J} _K}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^G}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} _d} ^H}} {{{{ e} ^e} _E}} {{{{ e} _e} ^K}} {{{{ e} ^f} _F}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _G} _H} _{,f}}} {{{{{ e} _c} ^J} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} ^D} ^E}} {{{{ \bar{\gamma}} ^F} ^I}} {{{{ \bar{\gamma}} _G} _H}} {{{{ \bar{\gamma}} _J} _K}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{ e} _e} ^H}} {{{{ e} ^f} _F}} {{{{ e} _f} ^K}} {{{{ e} ^i} _I}} {{{{{ e} _b} ^G} _{,c}}} {{{{{ e} _d} ^J} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} ^D} ^E}} {{{{ \bar{\gamma}} ^F} ^I}} {{{{ \bar{\gamma}} _G} _H}} {{{{ \bar{\gamma}} _J} _K}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^G}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{ e} ^f} _F}} {{{{ e} _f} ^K}} {{{{ e} ^i} _I}} {{{{{ e} _d} ^J} _{,a}}} {{{{{ e} _e} ^H} _{,c}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} ^D} ^E}} {{{{ \bar{\gamma}} ^F} ^I}} {{{{ \bar{\gamma}} _J} _K}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^G}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} _d} ^J}} {{{{ e} ^e} _E}} {{{{ e} _e} ^H}} {{{{ e} ^f} _F}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _G} _H} _{,c}}} {{{{{ e} _f} ^K} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} ^D} ^E}} {{{{ \bar{\gamma}} ^F} ^I}} {{{{ \bar{\gamma}} _G} _H}} {{{{ \bar{\gamma}} _J} _K}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^G}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} _d} ^J}} {{{{ e} ^e} _E}} {{{{ e} ^f} _F}} {{{{ e} ^i} _I}} {{{{{ e} _e} ^H} _{,c}}} {{{{{ e} _f} ^K} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} ^D} ^E}} {{{{ \bar{\gamma}} ^F} ^I}} {{{{ \bar{\gamma}} _G} _H}} {{{{ \bar{\gamma}} _J} _K}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} _d} ^J}} {{{{ e} ^e} _E}} {{{{ e} _e} ^H}} {{{{ e} ^f} _F}} {{{{ e} ^i} _I}} {{{{{ e} _b} ^G} _{,c}}} {{{{{ e} _f} ^K} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} ^D} ^E}} {{{{ \bar{\gamma}} ^F} ^I}} {{{{ \bar{\gamma}} _G} _H}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^G}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} _d} ^J}} {{{{ e} ^e} _E}} {{{{ e} ^f} _F}} {{{{ e} _f} ^K}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _J} _K} _{,a}}} {{{{{ e} _e} ^H} _{,c}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} ^D} ^E}} {{{{ \bar{\gamma}} ^F} ^I}} {{{{ \bar{\gamma}} _J} _K}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^G}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{ e} _e} ^H}} {{{{ e} ^f} _F}} {{{{ e} _f} ^K}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _G} _H} _{,c}}} {{{{{ e} _d} ^J} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ \bar{\gamma}} _G} _H}} {{{{ \bar{\gamma}} _J} _K}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} _d} ^H}} {{{{ e} ^e} _E}} {{{{ e} _e} ^K}} {{{{ e} ^f} _F}} {{{{ e} ^i} _I}} {{{{{ e} _b} ^G} _{,f}}} {{{{{ e} _c} ^J} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} ^D} ^E}} {{{{ \bar{\gamma}} ^F} ^I}} {{{{ \bar{\gamma}} _G} _H}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} _d} ^J}} {{{{ e} ^e} _E}} {{{{ e} _e} ^H}} {{{{ e} ^f} _F}} {{{{ e} _f} ^K}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _J} _K} _{,a}}} {{{{{ e} _b} ^G} _{,c}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ \bar{\gamma}} _G} _H}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^G}} {{{{ e} ^c} _C}} {{{{ e} _c} ^J}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{ e} _e} ^K}} {{{{ e} ^f} _F}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _J} _K} _{,a}}} {{{{{ e} _d} ^H} _{,f}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ \bar{\gamma}} _J} _K}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^G}} {{{{ e} ^c} _C}} {{{{ e} _c} ^J}} {{{{ e} ^d} _D}} {{{{ e} _d} ^H}} {{{{ e} ^e} _E}} {{{{ e} ^f} _F}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _G} _H} _{,f}}} {{{{{ e} _e} ^K} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ \bar{\gamma}} _G} _H}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _c} ^J}} {{{{ e} ^d} _D}} {{{{ e} _d} ^H}} {{{{ e} ^e} _E}} {{{{ e} _e} ^K}} {{{{ e} ^f} _F}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _J} _K} _{,a}}} {{{{{ e} _b} ^G} _{,f}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} ^D} ^E}} {{{{ \bar{\gamma}} ^F} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^G}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} _d} ^J}} {{{{ e} ^e} _E}} {{{{ e} _e} ^H}} {{{{ e} ^f} _F}} {{{{ e} _f} ^K}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _G} _H} _{,c}}} {{{{{ \bar{\gamma}} _J} _K} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ \bar{\gamma}} _G} _H}} {{{{ \bar{\gamma}} _J} _K}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^G}} {{{{ e} ^c} _C}} {{{{ e} _c} ^J}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{ e} ^f} _F}} {{{{ e} ^i} _I}} {{{{{ e} _d} ^H} _{,f}}} {{{{{ e} _e} ^K} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ \bar{\gamma}} _G} _H}} {{{{ \bar{\gamma}} _J} _K}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _c} ^J}} {{{{ e} ^d} _D}} {{{{ e} _d} ^H}} {{{{ e} ^e} _E}} {{{{ e} ^f} _F}} {{{{ e} ^i} _I}} {{{{{ e} _b} ^G} _{,f}}} {{{{{ e} _e} ^K} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ \bar{\gamma}} _G} _H}} {{{{ \bar{\gamma}} _J} _K}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^G}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{ e} _e} ^K}} {{{{ e} ^f} _F}} {{{{ e} ^i} _I}} {{{{{ e} _c} ^J} _{,a}}} {{{{{ e} _d} ^H} _{,f}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^G}} {{{{ e} ^c} _C}} {{{{ e} _c} ^J}} {{{{ e} ^d} _D}} {{{{ e} _d} ^H}} {{{{ e} ^e} _E}} {{{{ e} _e} ^K}} {{{{ e} ^f} _F}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _G} _H} _{,f}}} {{{{{ \bar{\gamma}} _J} _K} _{,a}}}}}$
cancelling forward and inverse transforms, and simplifying deltas...
${{{ \bar{\Lambda}} ^I} _{,t}} = {{{{\frac{1}{6}}} {{{ \beta} ^I}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{3}}} {{{ \beta} ^B}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ e} ^a} _A}} {{{{{{ e} ^b} _B} _{,a}} _{,b}}}} + {{{\frac{1}{3}}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{{ \beta} ^B} _{,a}} _{,b}}}} + {{{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ e} ^b} _B}} {{{{{ e} ^a} _A} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \beta} ^A} _{,b}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \bar{\gamma}} _{,b}}} {{{ \beta} ^B}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{\frac{1}{{\bar{\gamma}}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \bar{\gamma}} _{,b}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{\frac{1}{{\bar{\gamma}}^{2}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{ \beta} ^C}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{{ \hat{\Gamma}} ^I} _A} _B}} {{{{{ e} ^c} _C} _{,c}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \beta} ^C} _{,c}}} {{{{ e} ^c} _C}} {{{{{ \hat{\Gamma}} ^I} _A} _B}}} + {{{{{ \bar{\gamma}} ^A} ^B}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{{ \beta} ^I} _{,a}} _{,b}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^B}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ e} ^a} _A}} {{{{{ e} ^b} _B} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{{ \hat{\Gamma}} ^I} _B} _C}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \beta} ^B} _{,b}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{2}}} {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{\gamma}} ^B} ^A}} {{{{ \beta} ^I} _{,b}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{\frac{1}{\bar{\gamma}}}}} + {{{{ \beta} ^J}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _i} ^I}} {{{{{{ e} ^i} _J} _{,a}} _{,b}}}} + {{{\frac{1}{2}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^J}} {{{{ \bar{\gamma}} ^B} ^A}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _i} ^I}} {{{{{ e} ^i} _J} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _A} _B}} {{{{ \bar{\gamma}} ^C} ^A}} {{{{ \bar{\gamma}} ^D} ^B}} {{{{{ \hat{\Gamma}} ^I} _C} _D}}} + {{{-2}} {{{ \alpha} _{,a}}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^A} ^C}} {{{{ \bar{\gamma}} ^I} ^B}} {{{{ e} ^a} _A}}} + {{{2}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \beta} ^C} _{,a}}} {{{{ e} ^a} _A}} {{{{{ \hat{\Gamma}} ^I} _B} _C}}} + {{{-1}} {{\alpha}} \cdot {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} ^I} ^C}} {{{{ e} ^a} _A}} {{\frac{1}{\bar{\gamma}}}}} + {{{2}} {{{ \beta} ^C}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ e} ^a} _A}} {{{{ e} _c} ^D}} {{{{{ \hat{\Gamma}} ^I} _B} _D}} {{{{{ e} ^c} _C} _{,a}}}} + {{{\alpha}} \cdot {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{{ \bar{A}} _B} _C} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{{{ \bar{\gamma}} _C} _D} _{,a}} _{,b}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \beta} ^D} _{,b}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{{ \bar{\gamma}} _C} _D} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \beta} ^D} _{,c}}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{{ \bar{\gamma}} _B} _D} _{,a}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^c} _C}} {{{{{ \bar{A}} _A} _B} _{,c}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} ^C} ^I}} {{{{ e} ^b} _B}} {{{{{ \bar{\gamma}} _A} _C} _{,b}}} {{{{{ e} ^d} _D} _{,d}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} ^C} ^I}} {{{{ \beta} ^D} _{,d}}} {{{{ e} ^b} _B}} {{{{ e} ^d} _D}} {{{{{ \bar{\gamma}} _A} _C} _{,b}}}} + {{{\alpha}} \cdot {{{{ \bar{A}} _B} _E}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{{ e} _c} ^E} _{,a}}}} + {{{\alpha}} \cdot {{{{ \bar{A}} _D} _C}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{{ e} _b} ^D} _{,a}}}} + {{{-1}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^b} _B}} {{{{{ \bar{\gamma}} _A} _C} _{,d}}} {{{{{ e} ^d} _D} _{,b}}}} + {{{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} ^D} ^E}} {{{{ e} ^a} _A}} {{{{{ \bar{\gamma}} _B} _D} _{,a}}} {{{{{ \hat{\Gamma}} ^I} _C} _E}}} + {{{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^c} _C}} {{{{{ \bar{\gamma}} _A} _B} _{,d}}} {{{{{ e} ^d} _D} _{,c}}}} + {{{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ e} ^a} _A}} {{{{ e} ^d} _D}} {{{{{{ \bar{\gamma}} _B} _C} _{,a}} _{,d}}}} + {{{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \beta} ^D} _{,a}}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{{ \bar{\gamma}} _B} _D} _{,c}}}} + {{{-1}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \beta} ^I} _{,c}}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{{ \bar{\gamma}} _A} _D} _{,b}}}} + {{{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \beta} ^D} _{,c}}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{{ \bar{\gamma}} _A} _B} _{,d}}}} + {{{-1}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} ^C} ^I}} {{{{ \beta} ^D} _{,c}}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{{ \bar{\gamma}} _A} _D} _{,b}}}} + {{{-1}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \beta} ^D} _{,b}}} {{{{ e} ^b} _B}} {{{{ e} ^d} _D}} {{{{{ \bar{\gamma}} _A} _C} _{,d}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} ^D} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{{ \bar{\gamma}} _B} _D} _{,c}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{2}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^B}} {{{{ \bar{\gamma}} ^C} ^A}} {{{{ \bar{\gamma}} ^D} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{{ \bar{\gamma}} _C} _D} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{2}}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^b} _F}} {{{{{ e} ^d} _D} _{,b}}} {{{{{ e} _d} ^F} _{,a}}}} + {{{\frac{1}{2}}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _d} ^F}} {{{{{ \bar{\gamma}} _C} _F} _{,a}}} {{{{{ e} ^d} _D} _{,b}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{{{ e} _c} ^C} _{,a}} _{,b}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^d} _F}} {{{{{{ e} _d} ^F} _{,a}} _{,b}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^c} _F}} {{{{{ e} ^d} _D} _{,c}}} {{{{{ e} _d} ^F} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{ e} _d} ^F}} {{{{{ \bar{\gamma}} _B} _F} _{,a}}} {{{{{ e} ^d} _D} _{,c}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \beta} ^D} _{,c}}} {{{{ e} ^a} _A}} {{{{ e} ^c} _F}} {{{{ e} ^d} _D}} {{{{{ e} _d} ^F} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} _E} _D}} {{{{ \beta} ^D} _{,c}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{{ e} _b} ^E} _{,a}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} _E} _D}} {{{{ \beta} ^D} _{,b}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{{ e} _c} ^E} _{,a}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \beta} ^D} _{,b}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _F}} {{{{ e} ^d} _D}} {{{{{ e} _d} ^F} _{,a}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _A} _E}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{{ e} _b} ^E} _{,c}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _D} _B}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{{ e} _a} ^D} _{,c}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{{ e} _a} ^I} _{,b}}} {{{{{ e} ^d} _D} _{,d}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^C} ^I}} {{{{ e} ^b} _F}} {{{{ e} ^c} _C}} {{{{{ e} _c} ^F} _{,b}}} {{{{{ e} ^d} _D} _{,d}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^C} ^I}} {{{{ \beta} ^D} _{,d}}} {{{{ e} ^b} _F}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{{ e} _c} ^F} _{,b}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \beta} ^D} _{,d}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^d} _D}} {{{{{ e} _a} ^I} _{,b}}}} + {{{-1}} {{\alpha}} \cdot {{{{ \bar{A}} _A} _B}} {{{{ \bar{\gamma}} ^C} ^A}} {{{{ \bar{\gamma}} ^D} ^B}} {{{{ \bar{\gamma}} ^E} ^I}} {{{{ e} ^e} _E}} {{{{{ \bar{\gamma}} _C} _D} _{,e}}}} + {{{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{{ e} _a} ^C} _{,d}}} {{{{{ e} ^d} _D} _{,c}}}} + {{{-1}} {{{ \beta} ^J}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _i} ^I}} {{{{{ \bar{\gamma}} _A} _D} _{,b}}} {{{{{ e} ^i} _J} _{,c}}}} + {{{{ \beta} ^A}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{{{ e} _c} ^I} _{,a}} _{,d}}}} + {{{{ \beta} ^D}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{{ e} _b} ^I} _{,d}}} {{{{{ e} ^d} _D} _{,c}}}} + {{{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{{ \hat{\Gamma}} ^I} _C} _E}} {{{{{ e} _b} ^E} _{,a}}}} + {{{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{ e} _d} ^F}} {{{{{ \bar{\gamma}} _B} _F} _{,c}}} {{{{{ e} ^d} _D} _{,a}}}} + {{{-1}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^C} ^I}} {{{{ e} ^b} _F}} {{{{ e} ^c} _C}} {{{{{ e} ^d} _D} _{,c}}} {{{{{ e} _d} ^F} _{,b}}}} + {{{{ \beta} ^A}} {{{{ \bar{\gamma}} ^D} ^E}} {{{{ e} ^a} _A}} {{{{ e} ^d} _D}} {{{{{ \hat{\Gamma}} ^I} _G} _E}} {{{{{ e} _d} ^G} _{,a}}}} + {{{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^c} _F}} {{{{{ e} ^d} _D} _{,a}}} {{{{{ e} _d} ^F} _{,c}}}} + {{{-1}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{{ e} _c} ^I} _{,d}}} {{{{{ e} ^d} _D} _{,b}}}} + {{{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^d} _D}} {{{{{{ e} _b} ^D} _{,a}} _{,d}}}} + {{{-1}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} ^C} ^I}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _d} ^F}} {{{{{ \bar{\gamma}} _A} _F} _{,b}}} {{{{{ e} ^d} _D} _{,c}}}} + {{{-1}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{{ e} _a} ^B} _{,d}}} {{{{{ e} ^d} _D} _{,b}}}} + {{{-1}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \beta} ^I} _{,c}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{{ e} _a} ^C} _{,b}}}} + {{{{{ \bar{\gamma}} ^A} ^I}} {{{{ \beta} ^D} _{,a}}} {{{{ e} ^a} _A}} {{{{ e} ^c} _F}} {{{{ e} ^d} _D}} {{{{{ e} _d} ^F} _{,c}}}} + {{{-1}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} ^C} ^I}} {{{{ \bar{\gamma}} _E} _D}} {{{{ \beta} ^D} _{,c}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{{ e} _a} ^E} _{,b}}}} + {{{-1}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \beta} ^D} _{,b}}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{{ e} _c} ^I} _{,d}}}} + {{{-1}} {{{{ \bar{\gamma}} ^C} ^I}} {{{{ \beta} ^D} _{,c}}} {{{{ e} ^b} _F}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{{ e} _d} ^F} _{,b}}}} + {{{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} _E} _D}} {{{{ \beta} ^D} _{,a}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{{ e} _b} ^E} _{,c}}}} + {{{{{ \bar{\gamma}} ^B} ^C}} {{{{ \beta} ^D} _{,c}}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{{ e} _b} ^I} _{,d}}}} + {{{-1}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \beta} ^I} _{,c}}} {{{{ e} ^b} _F}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{{ e} _d} ^F} _{,b}}}} + {{{-1}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \beta} ^D} _{,b}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^d} _D}} {{{{{ e} _a} ^B} _{,d}}}} + {{{{{ \bar{\gamma}} ^A} ^I}} {{{{ \beta} ^D} _{,c}}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{{ e} _a} ^C} _{,d}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{{ e} _b} ^I} _{,c}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^D} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^c} _F}} {{{{ e} ^d} _D}} {{{{{ e} _d} ^F} _{,c}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{2}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^B}} {{{{ \bar{\gamma}} ^D} ^I}} {{{{ e} ^a} _F}} {{{{ e} ^b} _B}} {{{{ e} ^d} _D}} {{{{{ e} _d} ^F} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{2}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^B}} {{{{ \bar{\gamma}} ^C} ^A}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{{ e} _c} ^I} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{2}}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} _E} _F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _d} ^F}} {{{{{ e} _c} ^E} _{,a}}} {{{{{ e} ^d} _D} _{,b}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} _E} _F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _d} ^F}} {{{{{ e} _b} ^E} _{,a}}} {{{{{ e} ^d} _D} _{,c}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _A} _B}} {{{{ \bar{\gamma}} ^C} ^A}} {{{{ \bar{\gamma}} ^D} ^E}} {{{{ \bar{\gamma}} ^I} ^B}} {{{{ e} ^e} _E}} {{{{{ \bar{\gamma}} _C} _D} _{,e}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _A} _B}} {{{{ \bar{\gamma}} ^C} ^A}} {{{{ \bar{\gamma}} ^D} ^B}} {{{{ \bar{\gamma}} ^E} ^I}} {{{{ e} ^d} _D}} {{{{{ \bar{\gamma}} _C} _E} _{,d}}}} + {{{-1}} {{\alpha}} \cdot {{{{ \bar{A}} _G} _B}} {{{{ \bar{\gamma}} ^D} ^B}} {{{{ \bar{\gamma}} ^E} ^I}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{{ e} _d} ^G} _{,e}}}} + {{{-1}} {{\alpha}} \cdot {{{{ \bar{A}} _A} _B}} {{{{ \bar{\gamma}} ^C} ^A}} {{{{ \bar{\gamma}} ^E} ^I}} {{{{ e} ^c} _C}} {{{{ e} ^e} _E}} {{{{{ e} _c} ^B} _{,e}}}} + {{{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} _E} _F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _d} ^F}} {{{{{ e} _b} ^E} _{,c}}} {{{{{ e} ^d} _D} _{,a}}}} + {{{-1}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} ^C} ^I}} {{{{ \bar{\gamma}} _E} _F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _d} ^F}} {{{{{ e} _a} ^E} _{,b}}} {{{{{ e} ^d} _D} _{,c}}}} + {{{-1}} {{{ \beta} ^J}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _i} ^I}} {{{{{ e} _a} ^C} _{,b}}} {{{{{ e} ^i} _J} _{,c}}}} + {{{-1}} {{{ \beta} ^J}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ e} ^b} _F}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} _i} ^I}} {{{{{ e} _d} ^F} _{,b}}} {{{{{ e} ^i} _J} _{,c}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _G} _B}} {{{{ \bar{\gamma}} ^D} ^E}} {{{{ \bar{\gamma}} ^I} ^B}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{{ e} _d} ^G} _{,e}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _A} _B}} {{{{ \bar{\gamma}} ^C} ^A}} {{{{ \bar{\gamma}} ^D} ^B}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{{ e} _c} ^I} _{,d}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _A} _B}} {{{{ \bar{\gamma}} ^C} ^A}} {{{{ \bar{\gamma}} ^I} ^B}} {{{{ e} ^c} _C}} {{{{ e} ^e} _E}} {{{{{ e} _c} ^E} _{,e}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _G} _B}} {{{{ \bar{\gamma}} ^D} ^B}} {{{{ \bar{\gamma}} ^E} ^I}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{{ e} _e} ^G} _{,d}}}} + {{{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{{ \bar{\gamma}} _C} _E} _{,a}}} {{{{{ \bar{\gamma}} _D} _F} _{,b}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ e} ^a} _A}} {{{{ e} ^f} _F}} {{{{{ \bar{\gamma}} _B} _D} _{,f}}} {{{{{ \bar{\gamma}} _C} _E} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} ^D} ^E}} {{{{ \bar{\gamma}} ^F} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{{ \bar{\gamma}} _B} _E} _{,c}}} {{{{{ \bar{\gamma}} _D} _F} _{,a}}}} + {{{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^f} _F}} {{{{{ \bar{\gamma}} _K} _E} _{,a}}} {{{{{ e} _f} ^K} _{,b}}}} + {{{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _F}} {{{{ e} ^d} _D}} {{{{{ \bar{\gamma}} _C} _E} _{,a}}} {{{{{ e} _d} ^I} _{,b}}}} + {{{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{{ \bar{\gamma}} _D} _F} _{,b}}} {{{{{ e} _c} ^F} _{,a}}}} + {{{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^e} _E}} {{{{{ \bar{\gamma}} _H} _F} _{,b}}} {{{{{ e} _e} ^H} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} ^D} ^E}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ δ} ^K} _K}} {{{{{ \bar{\gamma}} _B} _E} _{,c}}} {{{{{ e} _d} ^I} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ e} ^a} _A}} {{{{ e} ^e} _E}} {{{{ e} ^f} _F}} {{{{{ \bar{\gamma}} _B} _K} _{,f}}} {{{{{ e} _e} ^K} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} ^F} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{{ \bar{\gamma}} _D} _F} _{,a}}} {{{{{ e} _b} ^D} _{,c}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{ e} ^f} _F}} {{{{{ \bar{\gamma}} _E} _D} _{,f}}} {{{{{ e} _c} ^I} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^f} _F}} {{{{{ \bar{\gamma}} _C} _E} _{,a}}} {{{{{ e} _b} ^C} _{,f}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^D} ^E}} {{{{ \bar{\gamma}} ^F} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^c} _H}} {{{{ e} ^e} _E}} {{{{{ \bar{\gamma}} _D} _F} _{,a}}} {{{{{ e} _e} ^H} _{,c}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ e} ^a} _A}} {{{{ e} ^d} _D}} {{{{ e} ^f} _F}} {{{{{ \bar{\gamma}} _C} _E} _{,a}}} {{{{{ e} _d} ^I} _{,f}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} ^F} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{ e} ^f} _F}} {{{{{ \bar{\gamma}} _B} _K} _{,c}}} {{{{{ e} _f} ^K} _{,a}}}} + {{{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ e} ^a} _A}} {{{{ e} ^b} _F}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{{ e} _c} ^F} _{,a}}} {{{{{ e} _d} ^I} _{,b}}}} + {{{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _F}} {{{{ e} ^d} _H}} {{{{ e} ^e} _E}} {{{{{ e} _d} ^I} _{,b}}} {{{{{ e} _e} ^H} _{,a}}}} + {{{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ \bar{\gamma}} _H} _K}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^e} _E}} {{{{ e} ^f} _F}} {{{{{ e} _e} ^H} _{,a}}} {{{{{ e} _f} ^K} _{,b}}}} + {{{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _K}} {{{{ e} ^f} _F}} {{{{{ e} _c} ^F} _{,a}}} {{{{{ e} _f} ^K} _{,b}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ δ} ^K} _K}} {{{{{ e} _b} ^D} _{,c}}} {{{{{ e} _d} ^I} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ e} ^a} _A}} {{{{ e} ^d} _K}} {{{{ e} ^e} _E}} {{{{ e} ^f} _F}} {{{{{ e} _d} ^I} _{,f}}} {{{{{ e} _e} ^K} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^D} ^E}} {{{{ e} ^a} _A}} {{{{ e} ^c} _H}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{ δ} ^K} _K}} {{{{{ e} _d} ^I} _{,a}}} {{{{{ e} _e} ^H} _{,c}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _E}} {{{{ e} ^c} _C}} {{{{ e} ^f} _F}} {{{{{ e} _b} ^C} _{,f}}} {{{{{ e} _c} ^I} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} ^f} _F}} {{{{{ e} _c} ^I} _{,a}}} {{{{{ e} _d} ^F} _{,f}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} ^F} ^I}} {{{{ \bar{\gamma}} _D} _K}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} ^f} _F}} {{{{{ e} _b} ^D} _{,c}}} {{{{{ e} _f} ^K} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^F} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^c} _H}} {{{{ e} ^e} _K}} {{{{ e} ^f} _F}} {{{{{ e} _e} ^H} _{,c}}} {{{{{ e} _f} ^K} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ \bar{\gamma}} _C} _K}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^e} _E}} {{{{ e} ^f} _F}} {{{{{ e} _b} ^C} _{,f}}} {{{{{ e} _e} ^K} _{,a}}}}}$

extrinsic curvature trace evolution:
${\left( {K} = {{{{{ K} _i} _j}} {{{{ \gamma} ^i} ^j}}}\right)} _{,t}$
simplifying
${{ K} _{,t}} = {{{{{{ K} _i} _j}} {{{{{ \gamma} ^i} ^j} _{,t}}}} + {{{{{ \gamma} ^i} ^j}} {{{{{ K} _i} _j} _{,t}}}}}$
using ${{{{ \gamma} ^i} ^j} _{,t}} = {-{{{{{ \gamma} ^l} ^i}} {{{{ \gamma} ^m} ^j}} {{{{{ \gamma} _l} _m} _{,t}}}}}$
${{ K} _{,t}} = {{{{{{ K} _i} _j}} \cdot {-{{{{{ \gamma} ^l} ^i}} {{{{ \gamma} ^m} ^j}} {{{{{ \gamma} _l} _m} _{,t}}}}}} + {{{{{ \gamma} ^i} ^j}} {{{{{ K} _i} _j} _{,t}}}}}$
using ${{{{ \gamma} _i} _j} _{,t}} = {{{{{ \beta} ^k}} {{{{{ \gamma} _i} _j} _{,k}}}} + {{{{{ \beta} ^k} _{,i}}} {{{{ \gamma} _j} _k}}} + {{{{{ \beta} ^k} _{,j}}} {{{{ \gamma} _i} _k}}}{-{{{2}} {{\alpha}} \cdot {{{{ K} _i} _j}}}}}$
${{ K} _{,t}} = {{{{{{ \gamma} ^i} ^j}} {{{{{ K} _i} _j} _{,t}}}}{-{{{{ \beta} ^a}} {{{{ K} _i} _j}} {{{{ \gamma} ^l} ^i}} {{{{ \gamma} ^m} ^j}} {{{{{ \gamma} _l} _m} _{,a}}}}}{-{{{{{ K} _i} _j}} {{{{ \beta} ^a} _{,l}}} {{{{ \gamma} ^l} ^i}} {{{{ \gamma} _m} _a}} {{{{ \gamma} ^m} ^j}}}}{-{{{{{ K} _i} _j}} {{{{ \beta} ^a} _{,m}}} {{{{ \gamma} _l} _a}} {{{{ \gamma} ^l} ^i}} {{{{ \gamma} ^m} ^j}}}} + {{{2}} {{\alpha}} \cdot {{{{ K} _i} _j}} {{{{ K} _l} _m}} {{{{ \gamma} ^l} ^i}} {{{{ \gamma} ^m} ^j}}}}$
using ${{{ K} _i} _j} = {{{{{{ \bar{A}} _i} _j}} {{\frac{1}{{W}^{2}}}}} + {{{\frac{1}{3}}} {{K}} {{{{ \bar{\gamma}} _i} _j}} {{\frac{1}{{W}^{2}}}}}}$ , ${{{ \gamma} ^i} ^j} = {{{{{ \bar{\gamma}} ^i} ^j}} {{{W}^{2}}}}$ , ${{{ \gamma} _i} _j} = {\frac{{{ \bar{\gamma}} _i} _j}{{W}^{2}}}$
${{ K} _{,t}} = {{\frac{1}{9}}{\left({{{{9}} {{{{ \bar{\gamma}} ^i} ^j}} {{{{{ K} _i} _j} _{,t}}} {{{W}^{2}}}}{-{{{9}} {{{{ \bar{A}} _i} _j}} {{{{ \bar{\gamma}} _l} _a}} {{{{ \bar{\gamma}} ^l} ^i}} {{{{ \bar{\gamma}} ^m} ^j}} {{{{ \beta} ^a} _{,m}}}}}{-{{{9}} {{{{ \bar{A}} _i} _j}} {{{{ \bar{\gamma}} ^l} ^i}} {{{{ \bar{\gamma}} _m} _a}} {{{{ \bar{\gamma}} ^m} ^j}} {{{{ \beta} ^a} _{,l}}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{\gamma}} _i} _j}} {{{{ \bar{\gamma}} ^l} ^i}} {{{{ \bar{\gamma}} _l} _m}} {{{{ \bar{\gamma}} ^m} ^j}} {{{K}^{2}}}} + {{{18}} {{\alpha}} \cdot {{{{ \bar{A}} _i} _j}} {{{{ \bar{A}} _l} _m}} {{{{ \bar{\gamma}} ^l} ^i}} {{{{ \bar{\gamma}} ^m} ^j}}}{-{{{3}} {{K}} {{{{ \bar{\gamma}} _i} _j}} {{{{ \bar{\gamma}} ^l} ^i}} {{{{ \bar{\gamma}} _m} _a}} {{{{ \bar{\gamma}} ^m} ^j}} {{{{ \beta} ^a} _{,l}}}}}{-{{{3}} {{K}} {{{{ \bar{\gamma}} _i} _j}} {{{{ \bar{\gamma}} _l} _a}} {{{{ \bar{\gamma}} ^l} ^i}} {{{{ \bar{\gamma}} ^m} ^j}} {{{{ \beta} ^a} _{,m}}}}}{-{{{9}} {{{ \beta} ^a}} {{{{ \bar{A}} _i} _j}} {{{{ \bar{\gamma}} ^l} ^i}} {{{{ \bar{\gamma}} ^m} ^j}} {{{{{ \gamma} _l} _m} _{,a}}} {{{W}^{2}}}}} + {{{6}} {{K}} {{\alpha}} \cdot {{{{ \bar{A}} _l} _m}} {{{{ \bar{\gamma}} _i} _j}} {{{{ \bar{\gamma}} ^l} ^i}} {{{{ \bar{\gamma}} ^m} ^j}}} + {{{6}} {{K}} {{\alpha}} \cdot {{{{ \bar{A}} _i} _j}} {{{{ \bar{\gamma}} ^l} ^i}} {{{{ \bar{\gamma}} _l} _m}} {{{{ \bar{\gamma}} ^m} ^j}}}{-{{{3}} {{K}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} _i} _j}} {{{{ \bar{\gamma}} ^l} ^i}} {{{{ \bar{\gamma}} ^m} ^j}} {{{{{ \gamma} _l} _m} _{,a}}} {{{W}^{2}}}}}}\right)}}$
simplifying bar metrics
${{ K} _{,t}} = {{{{{{ \bar{\gamma}} ^i} ^j}} {{{{{ K} _i} _j} _{,t}}} {{{W}^{2}}}} + {{{-1}} {{{{ \bar{A}} _i} ^m}} {{{{ \beta} ^i} _{,m}}}} + {{{-1}} {{{{ \bar{A}} ^l} _j}} {{{{ \beta} ^j} _{,l}}}} + {{{2}} \cdot {{\frac{1}{9}}} {{\alpha}} \cdot {{{{ δ} _j} ^j}} {{{K}^{2}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _i} _j}} {{{{ \bar{A}} ^i} ^j}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{K}} {{{{ \beta} ^j} _{,j}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{K}} {{{{ \beta} ^a} _{,a}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{A}} ^l} ^m}} {{{{{ \gamma} _l} _m} _{,a}}} {{{W}^{2}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{K}} {{\alpha}} \cdot {{{{ \bar{A}} _j} ^j}}} + {{{2}} \cdot {{\frac{1}{3}}} {{K}} {{\alpha}} \cdot {{{{ \bar{A}} _m} ^m}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{K}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^m} ^j}} {{{{{ \gamma} _j} _m} _{,a}}} {{{W}^{2}}}}}$
using ${{{ \bar{A}} ^i} _i} = {0}$
${{ K} _{,t}} = {{\frac{1}{9}}{\left({{-{{{3}} {{K}} {{{{ \beta} ^j} _{,j}}}}}{-{{{3}} {{K}} {{{{ \beta} ^a} _{,a}}}}}{-{{{9}} {{{{ \bar{A}} ^l} _j}} {{{{ \beta} ^j} _{,l}}}}}{-{{{9}} {{{{ \bar{A}} _i} ^m}} {{{{ \beta} ^i} _{,m}}}}} + {{{2}} {{\alpha}} \cdot {{{{ δ} _j} ^j}} {{{K}^{2}}}} + {{{9}} {{{{ \bar{\gamma}} ^i} ^j}} {{{{{ K} _i} _j} _{,t}}} {{{W}^{2}}}} + {{{18}} {{\alpha}} \cdot {{{{ \bar{A}} _i} _j}} {{{{ \bar{A}} ^i} ^j}}}{-{{{9}} {{{ \beta} ^a}} {{{{ \bar{A}} ^l} ^m}} {{{{{ \gamma} _l} _m} _{,a}}} {{{W}^{2}}}}}{-{{{3}} {{K}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^m} ^j}} {{{{{ \gamma} _j} _m} _{,a}}} {{{W}^{2}}}}}}\right)}}$
using ${{{ δ} _j} ^j} = {3}$
${{ K} _{,t}} = {{\frac{1}{3}}{\left({{-{{{K}} {{{{ \beta} ^j} _{,j}}}}}{-{{{K}} {{{{ \beta} ^a} _{,a}}}}} + {{{2}} {{\alpha}} \cdot {{{K}^{2}}}}{-{{{3}} {{{{ \bar{A}} _i} ^m}} {{{{ \beta} ^i} _{,m}}}}}{-{{{3}} {{{{ \bar{A}} ^l} _j}} {{{{ \beta} ^j} _{,l}}}}} + {{{3}} {{{{ \bar{\gamma}} ^i} ^j}} {{{{{ K} _i} _j} _{,t}}} {{{W}^{2}}}} + {{{6}} {{\alpha}} \cdot {{{{ \bar{A}} _i} _j}} {{{{ \bar{A}} ^i} ^j}}}{-{{{3}} {{{ \beta} ^a}} {{{{ \bar{A}} ^l} ^m}} {{{{{ \gamma} _l} _m} _{,a}}} {{{W}^{2}}}}}{-{{{K}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^m} ^j}} {{{{{ \gamma} _j} _m} _{,a}}} {{{W}^{2}}}}}}\right)}}$
symmetrizing ${{ \bar{\gamma}} _i} _j$
${{ K} _{,t}} = {{\frac{1}{3}}{\left({{-{{{K}} {{{{ \beta} ^j} _{,j}}}}}{-{{{K}} {{{{ \beta} ^a} _{,a}}}}} + {{{2}} {{\alpha}} \cdot {{{K}^{2}}}}{-{{{3}} {{{{ \bar{A}} _i} ^m}} {{{{ \beta} ^i} _{,m}}}}}{-{{{3}} {{{{ \bar{A}} ^l} _j}} {{{{ \beta} ^j} _{,l}}}}} + {{{3}} {{{{ \bar{\gamma}} ^i} ^j}} {{{{{ K} _i} _j} _{,t}}} {{{W}^{2}}}} + {{{6}} {{\alpha}} \cdot {{{{ \bar{A}} ^i} ^j}} {{{{ \bar{A}} _i} _j}}}{-{{{3}} {{{ \beta} ^a}} {{{{ \bar{A}} ^l} ^m}} {{{{{ \gamma} _l} _m} _{,a}}} {{{W}^{2}}}}}{-{{{K}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^j} ^m}} {{{{{ \gamma} _j} _m} _{,a}}} {{{W}^{2}}}}}}\right)}}$
using definition of ${{{ K} _i} _j} _{,t}$
${{ K} _{,t}} = {{\frac{1}{6}}{\left({{-{{{2}} {{K}} {{{{ \beta} ^j} _{,j}}}}}{-{{{2}} {{K}} {{{{ \beta} ^a} _{,a}}}}} + {{{4}} {{\alpha}} \cdot {{{K}^{2}}}}{-{{{6}} {{{{ \bar{A}} ^l} _j}} {{{{ \beta} ^j} _{,l}}}}}{-{{{6}} {{{{ \bar{A}} _i} ^m}} {{{{ \beta} ^i} _{,m}}}}}{-{{{6}} {{{{ \alpha} _{,i}} _{,j}}} {{{{ \bar{\gamma}} ^i} ^j}} {{{W}^{2}}}}} + {{{12}} {{\alpha}} \cdot {{{{ \bar{A}} ^i} ^j}} {{{{ \bar{A}} _i} _j}}} + {{{6}} {{\alpha}} \cdot {{{{ R} _i} _j}} {{{{ \bar{\gamma}} ^i} ^j}} {{{W}^{2}}}}{-{{{6}} {{{ \beta} ^a}} {{{{ \bar{A}} ^l} ^m}} {{{{{ \gamma} _l} _m} _{,a}}} {{{W}^{2}}}}} + {{{6}} {{{ \beta} ^b}} {{{{ \bar{\gamma}} ^i} ^j}} {{{{{ K} _i} _j} _{,b}}} {{{W}^{2}}}} + {{{6}} {{{{ K} _b} _i}} {{{{ \bar{\gamma}} ^i} ^j}} {{{{ \beta} ^b} _{,j}}} {{{W}^{2}}}} + {{{6}} {{{{ K} _b} _j}} {{{{ \bar{\gamma}} ^i} ^j}} {{{{ \beta} ^b} _{,i}}} {{{W}^{2}}}}{-{{{2}} {{K}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^j} ^m}} {{{{{ \gamma} _j} _m} _{,a}}} {{{W}^{2}}}}}{-{{{3}} {{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^i} ^j}} {{{{ \gamma} ^b} ^c}} {{{{{ \gamma} _i} _j} _{,c}}} {{{W}^{2}}}}} + {{{3}} {{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^i} ^j}} {{{{ \gamma} ^b} ^c}} {{{{{ \gamma} _c} _i} _{,j}}} {{{W}^{2}}}} + {{{3}} {{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^i} ^j}} {{{{ \gamma} ^b} ^c}} {{{{{ \gamma} _c} _j} _{,i}}} {{{W}^{2}}}} + {{{6}} {{K}} {{\alpha}} \cdot {{{{ K} _i} _j}} {{{{ \bar{\gamma}} ^i} ^j}} {{{W}^{2}}}}{-{{{48}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _i} _j}} {{{{ \bar{\gamma}} ^i} ^j}} {{{W}^{2}}}}}{-{{{12}} {{\alpha}} \cdot {{{{ K} _d} _j}} {{{{ K} _i} _b}} {{{{ \bar{\gamma}} ^i} ^j}} {{{{ \gamma} ^b} ^d}} {{{W}^{2}}}}} + {{{24}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ \bar{\gamma}} ^i} ^j}} {{{{ \gamma} _i} _j}} {{{W}^{2}}}}{-{{{24}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}} \cdot {{{{ \bar{\gamma}} ^i} ^j}} {{{{ \gamma} _i} _j}} {{{W}^{2}}}}}}\right)}}$
using ${{{ K} _i} _j} = {{{{{{ \bar{A}} _i} _j}} {{\frac{1}{{W}^{2}}}}} + {{{\frac{1}{3}}} {{K}} {{{{ \bar{\gamma}} _i} _j}} {{\frac{1}{{W}^{2}}}}}}$ , ${{{ \gamma} ^i} ^j} = {{{{{ \bar{\gamma}} ^i} ^j}} {{{W}^{2}}}}$ , ${{{ \gamma} _i} _j} = {\frac{{{ \bar{\gamma}} _i} _j}{{W}^{2}}}$
${{ K} _{,t}} = {{\frac{1}{6}}{\left({{-{{{2}} {{K}} {{{{ \beta} ^j} _{,j}}}}}{-{{{2}} {{K}} {{{{ \beta} ^a} _{,a}}}}} + {{{4}} {{\alpha}} \cdot {{{K}^{2}}}}{-{{{6}} {{{{ \bar{A}} ^l} _j}} {{{{ \beta} ^j} _{,l}}}}}{-{{{6}} {{{{ \bar{A}} _i} ^m}} {{{{ \beta} ^i} _{,m}}}}}{-{{{6}} {{{{ \alpha} _{,i}} _{,j}}} {{{{ \bar{\gamma}} ^i} ^j}} {{{W}^{2}}}}} + {{{12}} {{\alpha}} \cdot {{{{ \bar{A}} ^i} ^j}} {{{{ \bar{A}} _i} _j}}} + {{{6}} {{\alpha}} \cdot {{{{ R} _i} _j}} {{{{ \bar{\gamma}} ^i} ^j}} {{{W}^{2}}}}{-{{{6}} {{{ \beta} ^a}} {{{{ \bar{A}} ^l} ^m}} {{{{{ \gamma} _l} _m} _{,a}}} {{{W}^{2}}}}} + {{{6}} {{{ \beta} ^b}} {{{{ \bar{\gamma}} ^i} ^j}} {{{{{ K} _i} _j} _{,b}}} {{{W}^{2}}}} + {{{6}} {{\left({{{{{{ \bar{A}} _b} _i}} {{\frac{1}{{W}^{2}}}}} + {{{\frac{1}{3}}} {{K}} {{{{ \bar{\gamma}} _b} _i}} {{\frac{1}{{W}^{2}}}}}}\right)}} {{{{ \bar{\gamma}} ^i} ^j}} {{{{ \beta} ^b} _{,j}}} {{{W}^{2}}}} + {{{6}} {{\left({{{{{{ \bar{A}} _b} _j}} {{\frac{1}{{W}^{2}}}}} + {{{\frac{1}{3}}} {{K}} {{{{ \bar{\gamma}} _b} _j}} {{\frac{1}{{W}^{2}}}}}}\right)}} {{{{ \bar{\gamma}} ^i} ^j}} {{{{ \beta} ^b} _{,i}}} {{{W}^{2}}}}{-{{{2}} {{K}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^j} ^m}} {{{{{ \gamma} _j} _m} _{,a}}} {{{W}^{2}}}}}{-{{{3}} {{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^i} ^j}} {{{{{{ \bar{\gamma}} ^b} ^c}} {{{W}^{2}}}}} {{{{{ \gamma} _i} _j} _{,c}}} {{{W}^{2}}}}} + {{{3}} {{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^i} ^j}} {{{{{{ \bar{\gamma}} ^b} ^c}} {{{W}^{2}}}}} {{{{{ \gamma} _c} _i} _{,j}}} {{{W}^{2}}}} + {{{3}} {{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^i} ^j}} {{{{{{ \bar{\gamma}} ^b} ^c}} {{{W}^{2}}}}} {{{{{ \gamma} _c} _j} _{,i}}} {{{W}^{2}}}} + {{{6}} {{K}} {{\alpha}} \cdot {{\left({{{{{{ \bar{A}} _i} _j}} {{\frac{1}{{W}^{2}}}}} + {{{\frac{1}{3}}} {{K}} {{{{ \bar{\gamma}} _i} _j}} {{\frac{1}{{W}^{2}}}}}}\right)}} {{{{ \bar{\gamma}} ^i} ^j}} {{{W}^{2}}}}{-{{{48}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _i} _j}} {{{{ \bar{\gamma}} ^i} ^j}} {{{W}^{2}}}}}{-{{{12}} {{\alpha}} \cdot {{\left({{{{{{ \bar{A}} _d} _j}} {{\frac{1}{{W}^{2}}}}} + {{{\frac{1}{3}}} {{K}} {{{{ \bar{\gamma}} _d} _j}} {{\frac{1}{{W}^{2}}}}}}\right)}} {{\left({{{{{{ \bar{A}} _i} _b}} {{\frac{1}{{W}^{2}}}}} + {{{\frac{1}{3}}} {{K}} {{{{ \bar{\gamma}} _i} _b}} {{\frac{1}{{W}^{2}}}}}}\right)}} {{{{ \bar{\gamma}} ^i} ^j}} {{{{{{ \bar{\gamma}} ^b} ^d}} {{{W}^{2}}}}} {{{W}^{2}}}}} + {{{24}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ \bar{\gamma}} ^i} ^j}} {{\frac{{{ \bar{\gamma}} _i} _j}{{W}^{2}}}} {{{W}^{2}}}}{-{{{24}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}} \cdot {{{{ \bar{\gamma}} ^i} ^j}} {{\frac{{{ \bar{\gamma}} _i} _j}{{W}^{2}}}} {{{W}^{2}}}}}}\right)}}$
simplifying bar metrics
${{ K} _{,t}} = {{{{-1}} \cdot {{\frac{1}{3}}} {{K}} {{{{ \beta} ^a} _{,a}}}} + {{{\frac{1}{3}}} {{K}} {{{{ \beta} ^b} _{,b}}}} + {{{{{ \bar{A}} _b} ^i}} {{{{ \beta} ^b} _{,i}}}} + {{{-1}} {{{{ \bar{A}} _i} ^m}} {{{{ \beta} ^i} _{,m}}}} + {{{-1}} {{{{ \bar{A}} ^l} _j}} {{{{ \beta} ^j} _{,l}}}} + {{{{{ \bar{A}} _b} ^j}} {{{{ \beta} ^b} _{,j}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{\alpha}} \cdot {{{K}^{2}}}} + {{{-2}} \cdot {{\frac{1}{9}}} {{\alpha}} \cdot {{{{ δ} ^d} _d}} {{{K}^{2}}}} + {{{\frac{1}{3}}} {{K}} {{\alpha}} \cdot {{{{ \bar{A}} ^j} _j}}} + {{{\frac{1}{3}}} {{\alpha}} \cdot {{{{ δ} _j} ^j}} {{{K}^{2}}}} + {{{-1}} {{{{ \alpha} _{,i}} _{,j}}} {{{{ \bar{\gamma}} ^i} ^j}} {{{W}^{2}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{K}} {{\alpha}} \cdot {{{{ \bar{A}} _d} ^d}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _d} _j}} {{{{ \bar{A}} ^j} ^d}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _i} _j}} {{{{ \bar{A}} ^i} ^j}}} + {{{\alpha}} \cdot {{{{ R} _i} _j}} {{{{ \bar{\gamma}} ^i} ^j}} {{{W}^{2}}}} + {{{{ \beta} ^b}} {{{{ \bar{\gamma}} ^i} ^j}} {{{{{ K} _i} _j} _{,b}}} {{{W}^{2}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{A}} ^l} ^m}} {{{{{ \gamma} _l} _m} _{,a}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^i} ^j}} {{{{{ \gamma} _i} _j} _{,c}}} {{{W}^{4}}}} + {{{4}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ δ} ^j} _j}}} + {{{-4}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}} \cdot {{{{ δ} ^j} _j}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{K}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^j} ^m}} {{{{{ \gamma} _j} _m} _{,a}}} {{{W}^{2}}}} + {{{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^i} ^j}} {{{{{ \gamma} _c} _i} _{,j}}} {{{W}^{4}}}} + {{{-8}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _i} _j}} {{{{ \bar{\gamma}} ^i} ^j}} {{{W}^{2}}}}}$
using ${{{ \bar{A}} ^i} _i} = {0}$
${{ K} _{,t}} = {{{{-1}} \cdot {{\frac{1}{3}}} {{K}} {{{{ \beta} ^a} _{,a}}}} + {{{\frac{1}{3}}} {{K}} {{{{ \beta} ^b} _{,b}}}} + {{{-1}} {{{{ \bar{A}} _i} ^m}} {{{{ \beta} ^i} _{,m}}}} + {{{{{ \bar{A}} _b} ^i}} {{{{ \beta} ^b} _{,i}}}} + {{{-1}} {{{{ \bar{A}} _j} ^l}} {{{{ \beta} ^j} _{,l}}}} + {{{{{ \bar{A}} _b} ^j}} {{{{ \beta} ^b} _{,j}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{\alpha}} \cdot {{{K}^{2}}}} + {{{-2}} \cdot {{\frac{1}{9}}} {{\alpha}} \cdot {{{{ δ} ^d} _d}} {{{K}^{2}}}} + {{{\frac{1}{3}}} {{\alpha}} \cdot {{{{ δ} _j} ^j}} {{{K}^{2}}}} + {{{-1}} {{{{ \alpha} _{,i}} _{,j}}} {{{{ \bar{\gamma}} ^i} ^j}} {{{W}^{2}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} ^d} ^j}} {{{{ \bar{A}} _d} _j}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} ^i} ^j}} {{{{ \bar{A}} _i} _j}}} + {{{\alpha}} \cdot {{{{ R} _i} _j}} {{{{ \bar{\gamma}} ^i} ^j}} {{{W}^{2}}}} + {{{{ \beta} ^b}} {{{{ \bar{\gamma}} ^i} ^j}} {{{{{ K} _i} _j} _{,b}}} {{{W}^{2}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{A}} ^l} ^m}} {{{{{ \gamma} _l} _m} _{,a}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^i} ^j}} {{{{{ \gamma} _i} _j} _{,c}}} {{{W}^{4}}}} + {{{4}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ δ} ^j} _j}}} + {{{-4}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}} \cdot {{{{ δ} ^j} _j}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{K}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^j} ^m}} {{{{{ \gamma} _j} _m} _{,a}}} {{{W}^{2}}}} + {{{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^i} ^j}} {{{{{ \gamma} _c} _i} _{,j}}} {{{W}^{4}}}} + {{{-8}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _i} _j}} {{{{ \bar{\gamma}} ^i} ^j}} {{{W}^{2}}}}}$
using ${{{ δ} _j} ^j} = {3}$
${{ K} _{,t}} = {{{{-1}} \cdot {{\frac{1}{3}}} {{K}} {{{{ \beta} ^a} _{,a}}}} + {{{\frac{1}{3}}} {{K}} {{{{ \beta} ^b} _{,b}}}} + {{{\alpha}} \cdot {{{K}^{2}}}} + {{{-1}} {{{{ \bar{A}} _j} ^l}} {{{{ \beta} ^j} _{,l}}}} + {{{{{ \bar{A}} _b} ^j}} {{{{ \beta} ^b} _{,j}}}} + {{{{{ \bar{A}} _b} ^i}} {{{{ \beta} ^b} _{,i}}}} + {{{-1}} {{{{ \bar{A}} _i} ^m}} {{{{ \beta} ^i} _{,m}}}} + {{{-1}} {{{{ \alpha} _{,i}} _{,j}}} {{{{ \bar{\gamma}} ^i} ^j}} {{{W}^{2}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _i} _j}} {{{{ \bar{A}} ^i} ^j}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _d} _j}} {{{{ \bar{A}} ^d} ^j}}} + {{{12}} {{S}} {{\alpha}} \cdot {{\pi}}} + {{{-12}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}}} + {{{\alpha}} \cdot {{{{ R} _i} _j}} {{{{ \bar{\gamma}} ^i} ^j}} {{{W}^{2}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{A}} ^l} ^m}} {{{{{ \gamma} _l} _m} _{,a}}} {{{W}^{2}}}} + {{{{ \beta} ^b}} {{{{ \bar{\gamma}} ^i} ^j}} {{{{{ K} _i} _j} _{,b}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{K}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^j} ^m}} {{{{{ \gamma} _j} _m} _{,a}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^i} ^j}} {{{{{ \gamma} _i} _j} _{,c}}} {{{W}^{4}}}} + {{{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^i} ^j}} {{{{{ \gamma} _c} _i} _{,j}}} {{{W}^{4}}}} + {{{-8}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _i} _j}} {{{{ \bar{\gamma}} ^i} ^j}} {{{W}^{2}}}}}$
tidying indexes
${{ K} _{,t}} = {{{{\alpha}} \cdot {{{K}^{2}}}} + {{{-1}} {{{{ \alpha} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{12}} {{S}} {{\alpha}} \cdot {{\pi}}} + {{{-12}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}}} + {{{\alpha}} \cdot {{{{ R} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{A}} ^b} ^c}} {{{{{ \gamma} _b} _c} _{,a}}} {{{W}^{2}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ K} _b} _c} _{,a}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{K}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \gamma} _b} _c} _{,a}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \gamma} _c} _d} _{,b}}} {{{W}^{4}}}} + {{{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \gamma} _b} _c} _{,d}}} {{{W}^{4}}}} + {{{-8}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}}}$
using ${{{ K} _i} _j} = {{{{{{ \bar{A}} _i} _j}} {{\frac{1}{{W}^{2}}}}} + {{{\frac{1}{3}}} {{K}} {{{{ \bar{\gamma}} _i} _j}} {{\frac{1}{{W}^{2}}}}}}$
${{ K} _{,t}} = {{{{\alpha}} \cdot {{{K}^{2}}}} + {{{-1}} {{{{ \alpha} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{\alpha}} \cdot {{{{ R} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{A}} ^b} ^c}} {{{{{ \gamma} _b} _c} _{,a}}} {{{W}^{2}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{\frac{1}{3}}} {{K}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{\gamma}} _b} _c} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{K}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \gamma} _b} _c} _{,a}}} {{{W}^{2}}}} + {{{\frac{1}{3}}} {{{ K} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} _b} _c}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \gamma} _c} _d} _{,b}}} {{{W}^{4}}}} + {{{12}} {{S}} {{\alpha}} \cdot {{\pi}}} + {{{-12}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}}} + {{{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \gamma} _b} _c} _{,d}}} {{{W}^{4}}}} + {{{-2}} {{{ W} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^c}} {{\frac{1}{W}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{K}} {{{ W} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} _b} _c}} {{\frac{1}{W}}}} + {{{-8}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}}}$
simplifying bar metrics
${{ K} _{,t}} = {{{{\alpha}} \cdot {{{K}^{2}}}} + {{{-1}} {{{{ \alpha} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{\frac{1}{3}}} {{{ K} _{,a}}} {{{ \beta} ^a}} {{{{ δ} ^c} _c}}} + {{{-2}} {{{ W} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{A}} ^c} _c}} {{\frac{1}{W}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{K}} {{{ W} _{,a}}} {{{ \beta} ^a}} {{{{ δ} _c} ^c}} {{\frac{1}{W}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{\alpha}} \cdot {{{{ R} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{A}} ^b} ^c}} {{{{{ \gamma} _b} _c} _{,a}}} {{{W}^{2}}}} + {{{\frac{1}{3}}} {{K}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{\gamma}} _b} _c} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{K}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \gamma} _b} _c} _{,a}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \gamma} _c} _d} _{,b}}} {{{W}^{4}}}} + {{{12}} {{S}} {{\alpha}} \cdot {{\pi}}} + {{{-12}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}}} + {{{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \gamma} _b} _c} _{,d}}} {{{W}^{4}}}} + {{{-8}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}}}$
using ${{{ \bar{A}} ^i} _i} = {0}$
${{ K} _{,t}} = {{{{\alpha}} \cdot {{{K}^{2}}}} + {{{-1}} {{{{ \alpha} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{\frac{1}{3}}} {{{ K} _{,a}}} {{{ \beta} ^a}} {{{{ δ} ^c} _c}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{K}} {{{ W} _{,a}}} {{{ \beta} ^a}} {{{{ δ} _c} ^c}} {{\frac{1}{W}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{\alpha}} \cdot {{{{ R} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{A}} ^b} ^c}} {{{{{ \gamma} _b} _c} _{,a}}} {{{W}^{2}}}} + {{{\frac{1}{3}}} {{K}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{\gamma}} _b} _c} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{K}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \gamma} _b} _c} _{,a}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \gamma} _c} _d} _{,b}}} {{{W}^{4}}}} + {{{12}} {{S}} {{\alpha}} \cdot {{\pi}}} + {{{-12}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}}} + {{{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \gamma} _b} _c} _{,d}}} {{{W}^{4}}}} + {{{-8}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}}}$
using ${{{ δ} _j} ^j} = {3}$
${{ K} _{,t}} = {{{{\alpha}} \cdot {{{K}^{2}}}} + {{{{ K} _{,a}}} {{{ \beta} ^a}}} + {{{-1}} {{{{ \alpha} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{-2}} {{K}} {{{ W} _{,a}}} {{{ \beta} ^a}} {{\frac{1}{W}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{\alpha}} \cdot {{{{ R} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{A}} ^b} ^c}} {{{{{ \gamma} _b} _c} _{,a}}} {{{W}^{2}}}} + {{{\frac{1}{3}}} {{K}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{\gamma}} _b} _c} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{K}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \gamma} _b} _c} _{,a}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \gamma} _c} _d} _{,b}}} {{{W}^{4}}}} + {{{12}} {{S}} {{\alpha}} \cdot {{\pi}}} + {{{-12}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}}} + {{{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \gamma} _b} _c} _{,d}}} {{{W}^{4}}}} + {{{-8}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}}}$
using ${{{{ \Gamma} _i} _j} _k} = {{{\frac{1}{2}}} {{\left({{{{{ \gamma} _i} _j} _{,k}} + {{{{ \gamma} _i} _k} _{,j}}{-{{{{ \gamma} _j} _k} _{,i}}}}\right)}}}$ , ${{{{ \Gamma} ^i} _j} _k} = {{\frac{1}{2}} {{{{{ \gamma} ^i} ^m}} {{\left({{-{{{{ \gamma} _j} _k} _{,m}}} + {{{{ \gamma} _m} _k} _{,j}} + {{{{ \gamma} _m} _j} _{,k}}}\right)}}}}$ , ${{{ \gamma} ^i} ^j} = {{{{{ \bar{\gamma}} ^i} ^j}} {{{W}^{2}}}}$ , ${{{ \gamma} _i} _j} = {\frac{{{ \bar{\gamma}} _i} _j}{{W}^{2}}}$
${{ K} _{,t}} = {{{{\alpha}} \cdot {{{K}^{2}}}} + {{{{ K} _{,a}}} {{{ \beta} ^a}}} + {{{-1}} {{{{ \alpha} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{-2}} {{K}} {{{ W} _{,a}}} {{{ \beta} ^a}} {{\frac{1}{W}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{\alpha}} \cdot {{{{ R} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{A}} ^b} ^c}} {{{{{ \gamma} _b} _c} _{,a}}} {{{W}^{2}}}} + {{{\frac{1}{3}}} {{K}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{\gamma}} _b} _c} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{K}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \gamma} _b} _c} _{,a}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \gamma} _c} _d} _{,b}}} {{{W}^{4}}}} + {{{12}} {{S}} {{\alpha}} \cdot {{\pi}}} + {{{-12}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}}} + {{{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \gamma} _b} _c} _{,d}}} {{{W}^{4}}}} + {{{-8}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}}}$
simplifying bar metrics
${{ K} _{,t}} = {{{{\alpha}} \cdot {{{K}^{2}}}} + {{{{ K} _{,a}}} {{{ \beta} ^a}}} + {{{-1}} {{{{ \alpha} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{-2}} {{K}} {{{ W} _{,a}}} {{{ \beta} ^a}} {{\frac{1}{W}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{\alpha}} \cdot {{{{ R} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{A}} ^b} ^c}} {{{{{ \gamma} _b} _c} _{,a}}} {{{W}^{2}}}} + {{{\frac{1}{3}}} {{K}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{\gamma}} _b} _c} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{K}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \gamma} _b} _c} _{,a}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \gamma} _c} _d} _{,b}}} {{{W}^{4}}}} + {{{12}} {{S}} {{\alpha}} \cdot {{\pi}}} + {{{-12}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}}} + {{{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \gamma} _b} _c} _{,d}}} {{{W}^{4}}}} + {{{-8}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}}}$
using ${{{ \bar{A}} ^i} _i} = {0}$
${{ K} _{,t}} = {{{{\alpha}} \cdot {{{K}^{2}}}} + {{{{ K} _{,a}}} {{{ \beta} ^a}}} + {{{-1}} {{{{ \alpha} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{-2}} {{K}} {{{ W} _{,a}}} {{{ \beta} ^a}} {{\frac{1}{W}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{\alpha}} \cdot {{{{ R} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{A}} ^b} ^c}} {{{{{ \gamma} _b} _c} _{,a}}} {{{W}^{2}}}} + {{{\frac{1}{3}}} {{K}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{\gamma}} _b} _c} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{K}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \gamma} _b} _c} _{,a}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \gamma} _c} _d} _{,b}}} {{{W}^{4}}}} + {{{12}} {{S}} {{\alpha}} \cdot {{\pi}}} + {{{-12}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}}} + {{{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \gamma} _b} _c} _{,d}}} {{{W}^{4}}}} + {{{-8}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}}}$
using ${{{ δ} _j} ^j} = {3}$
${{ K} _{,t}} = {{{{\alpha}} \cdot {{{K}^{2}}}} + {{{{ K} _{,a}}} {{{ \beta} ^a}}} + {{{-1}} {{{{ \alpha} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{-2}} {{K}} {{{ W} _{,a}}} {{{ \beta} ^a}} {{\frac{1}{W}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{\alpha}} \cdot {{{{ R} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{A}} ^b} ^c}} {{{{{ \gamma} _b} _c} _{,a}}} {{{W}^{2}}}} + {{{\frac{1}{3}}} {{K}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{\gamma}} _b} _c} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{K}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \gamma} _b} _c} _{,a}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \gamma} _c} _d} _{,b}}} {{{W}^{4}}}} + {{{12}} {{S}} {{\alpha}} \cdot {{\pi}}} + {{{-12}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}}} + {{{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \gamma} _b} _c} _{,d}}} {{{W}^{4}}}} + {{{-8}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}}}$
using ${{{{ \gamma} _i} _j} _{,k}} = {\frac{{{{W}} {{{{{ \bar{\gamma}} _i} _j} _{,k}}}}{-{{{2}} {{{ W} _{,k}}} {{{{ \bar{\gamma}} _i} _j}}}}}{{W}^{3}}}$ , ${{{{ \bar{\gamma}} _i} _j} _{,k}} = {{{{{ \bar{\Gamma}} _i} _k} _j} + {{{{ \bar{\Gamma}} _j} _k} _i}}$
${{ K} _{,t}} = {{{{\alpha}} \cdot {{{K}^{2}}}} + {{{{ K} _{,a}}} {{{ \beta} ^a}}} + {{{-1}} {{{{ \alpha} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{-2}} {{K}} {{{ W} _{,a}}} {{{ \beta} ^a}} {{\frac{1}{W}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{A}} ^b} ^c}} {{{{{ \bar{\Gamma}} _b} _a} _c}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{A}} ^b} ^c}} {{{{{ \bar{\Gamma}} _c} _a} _b}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{\alpha}} \cdot {{{{ R} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \bar{\Gamma}} _c} _b} _d}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \bar{\Gamma}} _d} _b} _c}} {{{W}^{2}}}} + {{{2}} {{{ W} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{A}} ^b} ^c}} {{{{ \bar{\gamma}} _b} _c}} {{\frac{1}{W}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{K}} {{{ W} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^c}} {{\frac{1}{W}}}} + {{{12}} {{S}} {{\alpha}} \cdot {{\pi}}} + {{{-12}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}}} + {{{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \bar{\Gamma}} _c} _d} _b}} {{{W}^{2}}}} + {{{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{ \bar{\Gamma}} _b} _d} _c}} {{{W}^{2}}}} + {{{-8}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{W}} {{{ W} _{,b}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} _c} _d}}} + {{{-2}} {{W}} {{{ W} _{,d}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} _b} _c}} {{{{ \bar{\gamma}} ^c} ^d}}}}$
simplifying bar metrics
${{ K} _{,t}} = {{{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{{ \bar{\Gamma}} _d} ^a} ^d}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{{ \bar{\Gamma}} ^d} ^a} _d}} {{{W}^{2}}}} + {{{\alpha}} \cdot {{{K}^{2}}}} + {{{{ K} _{,a}}} {{{ \beta} ^a}}} + {{{{ \alpha} _{,a}}} {{{{{ \bar{\Gamma}} ^d} _d} ^a}} {{{W}^{2}}}} + {{{{ \alpha} _{,a}}} {{{{{ \bar{\Gamma}} ^a} _d} ^d}} {{{W}^{2}}}} + {{{-1}} {{{{ \alpha} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{-2}} {{K}} {{{ W} _{,a}}} {{{ \beta} ^a}} {{\frac{1}{W}}}} + {{{2}} {{{ W} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{A}} _c} ^c}} {{\frac{1}{W}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{K}} {{{ W} _{,a}}} {{{ \beta} ^a}} {{{{ δ} _c} ^c}} {{\frac{1}{W}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{A}} ^b} ^c}} {{{{{ \bar{\Gamma}} _c} _a} _b}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{A}} ^b} ^c}} {{{{{ \bar{\Gamma}} _b} _a} _c}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{\alpha}} \cdot {{{{ R} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{-2}} {{W}} {{{ W} _{,d}}} {{{ \alpha} _{,c}}} {{{{ \bar{\gamma}} ^c} ^d}}} + {{{12}} {{S}} {{\alpha}} \cdot {{\pi}}} + {{{-12}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}}} + {{{W}} {{{ W} _{,b}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ δ} ^d} _d}}} + {{{-8}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}}}$
using ${{{ \bar{A}} ^i} _i} = {0}$
${{ K} _{,t}} = {{{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{{ \bar{\Gamma}} _d} ^a} ^d}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{{ \bar{\Gamma}} ^d} ^a} _d}} {{{W}^{2}}}} + {{{\alpha}} \cdot {{{K}^{2}}}} + {{{{ K} _{,a}}} {{{ \beta} ^a}}} + {{{{ \alpha} _{,a}}} {{{{{ \bar{\Gamma}} ^a} _d} ^d}} {{{W}^{2}}}} + {{{{ \alpha} _{,a}}} {{{{{ \bar{\Gamma}} ^d} _d} ^a}} {{{W}^{2}}}} + {{{-1}} {{{{ \alpha} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{-2}} {{K}} {{{ W} _{,a}}} {{{ \beta} ^a}} {{\frac{1}{W}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{K}} {{{ W} _{,a}}} {{{ \beta} ^a}} {{{{ δ} _c} ^c}} {{\frac{1}{W}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{\alpha}} \cdot {{{{ R} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{-2}} {{{ \beta} ^a}} {{{{ \bar{A}} ^b} ^c}} {{{{{ \bar{\Gamma}} _b} _a} _c}}} + {{{-2}} {{W}} {{{ W} _{,d}}} {{{ \alpha} _{,c}}} {{{{ \bar{\gamma}} ^c} ^d}}} + {{{12}} {{S}} {{\alpha}} \cdot {{\pi}}} + {{{-12}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}}} + {{{W}} {{{ W} _{,b}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ δ} ^d} _d}}} + {{{-8}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}}}$
using ${{{ δ} _j} ^j} = {3}$
${{ K} _{,t}} = {{{{\alpha}} \cdot {{{K}^{2}}}} + {{{{ K} _{,a}}} {{{ \beta} ^a}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{{ \bar{\Gamma}} _d} ^a} ^d}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{{ \bar{\Gamma}} ^d} ^a} _d}} {{{W}^{2}}}} + {{{{ \alpha} _{,a}}} {{{{{ \bar{\Gamma}} ^a} _d} ^d}} {{{W}^{2}}}} + {{{{ \alpha} _{,a}}} {{{{{ \bar{\Gamma}} ^d} _d} ^a}} {{{W}^{2}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{-1}} {{{{ \alpha} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{-2}} {{{ \beta} ^a}} {{{{ \bar{A}} ^b} ^c}} {{{{{ \bar{\Gamma}} _b} _a} _c}}} + {{{\alpha}} \cdot {{{{ R} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{-2}} {{W}} {{{ W} _{,d}}} {{{ \alpha} _{,c}}} {{{{ \bar{\gamma}} ^c} ^d}}} + {{{12}} {{S}} {{\alpha}} \cdot {{\pi}}} + {{{-12}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}}} + {{{3}} {{W}} {{{ W} _{,b}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}}} + {{{-8}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}}}$
symmetrizing ${{{ \bar{\Gamma}} ^i} _j} _k$
${{ K} _{,t}} = {{{{\alpha}} \cdot {{{K}^{2}}}} + {{{{ K} _{,a}}} {{{ \beta} ^a}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{{ \bar{\Gamma}} ^b} ^a} _b}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{{ \bar{\Gamma}} _b} ^a} ^b}} {{{W}^{2}}}} + {{{{ \alpha} _{,a}}} {{{{{ \bar{\Gamma}} ^b} ^a} _b}} {{{W}^{2}}}} + {{{{ \alpha} _{,a}}} {{{{{ \bar{\Gamma}} ^a} ^b} _b}} {{{W}^{2}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{-1}} {{{{ \alpha} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{-2}} {{{ \beta} ^a}} {{{{ \bar{A}} ^b} ^c}} {{{{{ \bar{\Gamma}} _b} _a} _c}}} + {{{\alpha}} \cdot {{{{ R} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{-2}} {{W}} {{{ W} _{,a}}} {{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^b} ^a}}} + {{{12}} {{S}} {{\alpha}} \cdot {{\pi}}} + {{{-12}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}}} + {{{3}} {{W}} {{{ W} _{,a}}} {{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^b} ^a}}} + {{{-8}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}}}$
using ${{{{{ S} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}}} = {\frac{S}{{W}^{2}}}$
${{ K} _{,t}} = {{{{\alpha}} \cdot {{{K}^{2}}}} + {{{{ K} _{,a}}} {{{ \beta} ^a}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{{ \bar{\Gamma}} ^b} ^a} _b}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{{ \bar{\Gamma}} _b} ^a} ^b}} {{{W}^{2}}}} + {{{{ \alpha} _{,a}}} {{{{{ \bar{\Gamma}} ^b} ^a} _b}} {{{W}^{2}}}} + {{{{ \alpha} _{,a}}} {{{{{ \bar{\Gamma}} ^a} ^b} _b}} {{{W}^{2}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{-1}} {{{{ \alpha} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{-2}} {{{ \beta} ^a}} {{{{ \bar{A}} ^b} ^c}} {{{{{ \bar{\Gamma}} _b} _a} _c}}} + {{{\alpha}} \cdot {{{{ R} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{-2}} {{W}} {{{ W} _{,a}}} {{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^b} ^a}}} + {{{12}} {{S}} {{\alpha}} \cdot {{\pi}}} + {{{-12}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}}} + {{{3}} {{W}} {{{ W} _{,a}}} {{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^b} ^a}}} + {{{-8}} {{\alpha}} \cdot {{\pi}} \cdot {{{W}^{2}}} {{\frac{S}{{W}^{2}}}}}}$
factoring out ${{ \bar{\gamma}} ^i} ^j$ to form ${{ \bar{A}} _i} _j$
${{ K} _{,t}} = {{{{\alpha}} \cdot {{{K}^{2}}}} + {{{{ K} _{,a}}} {{{ \beta} ^a}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{{ \bar{\Gamma}} ^b} ^a} _b}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{{ \bar{\Gamma}} _b} ^a} ^b}} {{{W}^{2}}}} + {{{{ \alpha} _{,a}}} {{{{{ \bar{\Gamma}} ^b} ^a} _b}} {{{W}^{2}}}} + {{{{ \alpha} _{,a}}} {{{{{ \bar{\Gamma}} ^a} ^b} _b}} {{{W}^{2}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{-1}} {{{{ \alpha} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{-2}} {{{ \beta} ^a}} {{{{{{ \bar{A}} _d} _e}} {{{{ \bar{\gamma}} ^b} ^d}} {{{{ \bar{\gamma}} ^c} ^e}}}} {{{{{ \bar{\Gamma}} _b} _a} _c}}} + {{{\alpha}} \cdot {{{{ R} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{-2}} {{W}} {{{ W} _{,a}}} {{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^b} ^a}}} + {{{12}} {{S}} {{\alpha}} \cdot {{\pi}}} + {{{-12}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}}} + {{{3}} {{W}} {{{ W} _{,a}}} {{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^b} ^a}}} + {{{-8}} {{\alpha}} \cdot {{\pi}} \cdot {{{W}^{2}}} {{\frac{S}{{W}^{2}}}}}}$
using ${{{{ \bar{\Gamma}} ^a} ^b} _b} = {{{ \bar{\Lambda}} ^a}{-{{ \mathcal{C}} ^a}} + {{ \hat{\Gamma}} ^a}}$
${{ K} _{,t}} = {{{{\alpha}} \cdot {{{K}^{2}}}} + {{{{ K} _{,a}}} {{{ \beta} ^a}}} + {{{{ \alpha} _{,a}}} {{{ \bar{\Lambda}} ^a}} {{{W}^{2}}}} + {{{{ \alpha} _{,a}}} {{{ \hat{\Gamma}} ^a}} {{{W}^{2}}}} + {{{-1}} {{{ \alpha} _{,a}}} {{{ \mathcal{C}} ^a}} {{{W}^{2}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{-1}} {{{{ \alpha} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{W}} {{{ W} _{,a}}} {{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}}} + {{{\alpha}} \cdot {{{{ R} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{4}} {{S}} {{\alpha}} \cdot {{\pi}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{A}} _d} _e}} {{{{ \bar{\gamma}} ^b} ^d}} {{{{ \bar{\gamma}} ^c} ^e}} {{{{{ \bar{\gamma}} _a} _b} _{,c}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{A}} _d} _e}} {{{{ \bar{\gamma}} ^b} ^d}} {{{{ \bar{\gamma}} ^c} ^e}} {{{{{ \bar{\gamma}} _b} _c} _{,a}}}} + {{{{ \beta} ^a}} {{{{ \bar{A}} _d} _e}} {{{{ \bar{\gamma}} ^b} ^d}} {{{{ \bar{\gamma}} ^c} ^e}} {{{{{ \bar{\gamma}} _a} _c} _{,b}}}} + {{{-12}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}}}}$
symmetrizing ${{{ \bar{\Gamma}} ^i} _j} _k$
${{ K} _{,t}} = {{{{\alpha}} \cdot {{{K}^{2}}}} + {{{{ K} _{,a}}} {{{ \beta} ^a}}} + {{{{ \alpha} _{,a}}} {{{ \bar{\Lambda}} ^a}} {{{W}^{2}}}} + {{{{ \alpha} _{,a}}} {{{ \hat{\Gamma}} ^a}} {{{W}^{2}}}} + {{{-1}} {{{ \alpha} _{,a}}} {{{ \mathcal{C}} ^a}} {{{W}^{2}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{-1}} {{{{ \alpha} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{W}} {{{ W} _{,a}}} {{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}}} + {{{\alpha}} \cdot {{{{ R} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{4}} {{S}} {{\alpha}} \cdot {{\pi}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^d}} {{{{ \bar{\gamma}} ^c} ^e}} {{{{{ \bar{\gamma}} _d} _e} _{,a}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^d}} {{{{ \bar{\gamma}} ^c} ^e}} {{{{{ \bar{\gamma}} _a} _d} _{,e}}}} + {{{{ \beta} ^a}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^d}} {{{{ \bar{\gamma}} ^c} ^e}} {{{{{ \bar{\gamma}} _a} _e} _{,d}}}} + {{{-12}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}}}}$

using locally-Minkowski normalized coordinates:
${{ K} _{,t}} = {{{{\alpha}} \cdot {{{K}^{2}}}} + {{{{ K} _{,a}}} {{{ \beta} ^A}} {{{{ e} ^a} _A}}} + {{{\alpha}} \cdot {{{{ R} _A} _B}} {{{{ \bar{\gamma}} ^A} ^B}} {{{W}^{2}}}} + {{{{ \alpha} _{,a}}} {{{ \bar{\Lambda}} ^A}} {{{{ e} ^a} _A}} {{{W}^{2}}}} + {{{{ \alpha} _{,a}}} {{{ \hat{\Gamma}} ^A}} {{{{ e} ^a} _A}} {{{W}^{2}}}} + {{{-1}} {{{ \alpha} _{,a}}} {{{ \mathcal{C}} ^A}} {{{{ e} ^a} _A}} {{{W}^{2}}}} + {{{4}} {{S}} {{\alpha}} \cdot {{\pi}}} + {{{-1}} {{{{ \alpha} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{W}^{2}}}} + {{{-12}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}}} + {{{W}} {{{ W} _{,a}}} {{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}}} + {{{{ \beta} ^A}} {{{{ \bar{A}} _D} _E}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^D}} {{{{ e} ^c} _C}} {{{{{ e} _c} ^E} _{,a}}}} + {{{{ \beta} ^A}} {{{{ \bar{A}} _D} _E}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _c} ^E}} {{{{{ e} _b} ^D} _{,a}}}} + {{{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^D}} {{{{ e} ^c} _C}} {{{{ e} _c} ^E}} {{{{{ \bar{A}} _D} _E} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _F} _G}} {{{{ e} ^a} _A}} {{{{ e} ^d} _D}} {{{{ e} _d} ^G}} {{{{ e} ^e} _E}} {{{{{ e} _a} ^F} _{,e}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _F} _G}} {{{{ e} ^a} _A}} {{{{ e} ^d} _D}} {{{{ e} _d} ^F}} {{{{ e} ^e} _E}} {{{{{ e} _e} ^G} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _F} _G}} {{{{ e} ^a} _A}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{ e} _e} ^G}} {{{{{ e} _d} ^F} _{,a}}}} + {{{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _F} _G}} {{{{ e} ^a} _A}} {{{{ e} _a} ^F}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{{ e} _e} ^G} _{,d}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _F} _G}} {{{{ e} ^a} _A}} {{{{ e} _a} ^F}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{{ e} _d} ^G} _{,e}}}} + {{{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _F} _G}} {{{{ e} ^a} _A}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{ e} _e} ^G}} {{{{{ e} _a} ^F} _{,d}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ e} ^a} _A}} {{{{ e} _a} ^F}} {{{{ e} ^d} _D}} {{{{ e} _d} ^G}} {{{{ e} ^e} _E}} {{{{{ \bar{\gamma}} _F} _G} _{,e}}}} + {{{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ e} ^a} _A}} {{{{ e} _a} ^F}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{ e} _e} ^G}} {{{{{ \bar{\gamma}} _F} _G} _{,d}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ e} ^a} _A}} {{{{ e} ^d} _D}} {{{{ e} _d} ^F}} {{{{ e} ^e} _E}} {{{{ e} _e} ^G}} {{{{{ \bar{\gamma}} _F} _G} _{,a}}}}}$
cancelling forward and inverse transforms, and simplifying deltas...
${{ K} _{,t}} = {{{{\alpha}} \cdot {{{K}^{2}}}} + {{{{ K} _{,a}}} {{{ \beta} ^A}} {{{{ e} ^a} _A}}} + {{{\alpha}} \cdot {{{{ R} _A} _B}} {{{{ \bar{\gamma}} ^A} ^B}} {{{W}^{2}}}} + {{{{ \alpha} _{,a}}} {{{ \bar{\Lambda}} ^A}} {{{{ e} ^a} _A}} {{{W}^{2}}}} + {{{{ \alpha} _{,a}}} {{{ \hat{\Gamma}} ^A}} {{{{ e} ^a} _A}} {{{W}^{2}}}} + {{{-1}} {{{ \alpha} _{,a}}} {{{ \mathcal{C}} ^A}} {{{{ e} ^a} _A}} {{{W}^{2}}}} + {{{4}} {{S}} {{\alpha}} \cdot {{\pi}}} + {{{-1}} {{{{ \alpha} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{W}^{2}}}} + {{{-12}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}}} + {{{W}} {{{ W} _{,a}}} {{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}}} + {{{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{{ \bar{A}} _B} _C} _{,a}}}} + {{{{ \beta} ^A}} {{{{ \bar{A}} _D} _C}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{{ e} _b} ^D} _{,a}}}} + {{{{ \beta} ^A}} {{{{ \bar{A}} _B} _E}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{{ e} _c} ^E} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ e} ^a} _A}} {{{{{ \bar{\gamma}} _D} _E} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ e} ^e} _E}} {{{{{ \bar{\gamma}} _A} _D} _{,e}}}} + {{{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ e} ^d} _D}} {{{{{ \bar{\gamma}} _A} _E} _{,d}}}} + {{{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ e} ^a} _A}} {{{{ e} ^d} _D}} {{{{{ e} _a} ^C} _{,d}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _A} _G}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{{ e} _d} ^G} _{,e}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ e} ^a} _A}} {{{{ e} ^d} _D}} {{{{{ e} _d} ^C} _{,a}}}} + {{{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _A} _G}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{{ e} _e} ^G} _{,d}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{A}} _G} _C}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ e} ^a} _A}} {{{{ e} ^e} _E}} {{{{{ e} _e} ^G} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ e} ^a} _A}} {{{{ e} ^e} _E}} {{{{{ e} _a} ^B} _{,e}}}}}$

trace-free extrinsic curvature evolution:
${{{{ A} _i} _j} _{,t}} = {{\frac{1}{3}}{\left({{{{3}} {{{{{ K} _i} _j} _{,t}}}}{-{{{K}} {{{{{ \gamma} _i} _j} _{,t}}}}}{-{{{{ K} _{,t}}} {{{{ \gamma} _i} _j}}}}}\right)}}$

conformal trace-free extrinsic curvature evolution:
${{{{ \bar{A}} _i} _j} _{,t}} = {{{W}} {{\left({{{{W}} {{{{{ A} _i} _j} _{,t}}}} + {{{2}} {{{ W} _{,t}}} {{{{ A} _i} _j}}}}\right)}}}$
using ${{ W} _{,t}} = {{{{-1}} \cdot {{\frac{1}{6}}} {{W}} {{{ \bar{\gamma}} _{,i}}} {{{ \beta} ^i}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{W}} {{{{ \beta} ^i} _{,i}}}} + {{{{ W} _{,i}}} {{{ \beta} ^i}}} + {{{\frac{1}{3}}} {{K}} {{W}} {{\alpha}}}}$
${{{{ \bar{A}} _i} _j} _{,t}} = {{{W}} {{\left({{{{W}} {{{{{ A} _i} _j} _{,t}}}} + {{{2}} {{\left({{{{-1}} \cdot {{\frac{1}{6}}} {{W}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{W}} {{{{ \beta} ^a} _{,a}}}} + {{{{ W} _{,a}}} {{{ \beta} ^a}}} + {{{\frac{1}{3}}} {{K}} {{W}} {{\alpha}}}}\right)}} {{{{ A} _i} _j}}}}\right)}}}$
using ${{{{ A} _i} _j} _{,t}} = {{\frac{1}{3}}{\left({{{{3}} {{{{{ K} _i} _j} _{,t}}}}{-{{{K}} {{{{{ \gamma} _i} _j} _{,t}}}}}{-{{{{ K} _{,t}}} {{{{ \gamma} _i} _j}}}}}\right)}}$
${{{{ \bar{A}} _i} _j} _{,t}} = {{{W}} {{\left({{{{W}} {{{\frac{1}{3}}{\left({{{{3}} {{{{{ K} _i} _j} _{,t}}}}{-{{{K}} {{{{{ \gamma} _i} _j} _{,t}}}}}{-{{{{ K} _{,t}}} {{{{ \gamma} _i} _j}}}}}\right)}}}} + {{{2}} {{\left({{{{-1}} \cdot {{\frac{1}{6}}} {{W}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{W}} {{{{ \beta} ^a} _{,a}}}} + {{{{ W} _{,a}}} {{{ \beta} ^a}}} + {{{\frac{1}{3}}} {{K}} {{W}} {{\alpha}}}}\right)}} {{{{ A} _i} _j}}}}\right)}}}$
using ${{{{ K} _i} _j} _{,t}} = {{{{-1}} {{{{ \alpha} _{,i}} _{,j}}}} + {{{\alpha}} \cdot {{{{ R} _i} _j}}} + {{{{ \beta} ^k}} {{{{{ K} _i} _j} _{,k}}}} + {{{{{ K} _k} _i}} {{{{ \beta} ^k} _{,j}}}} + {{{{{ K} _k} _j}} {{{{ \beta} ^k} _{,i}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,k}}} {{{{ \gamma} ^k} ^m}} {{{{{ \gamma} _i} _j} _{,m}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,k}}} {{{{ \gamma} ^k} ^m}} {{{{{ \gamma} _m} _j} _{,i}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,k}}} {{{{ \gamma} ^k} ^m}} {{{{{ \gamma} _m} _i} _{,j}}}} + {{{K}} {{\alpha}} \cdot {{{{ K} _i} _j}}} + {{{-8}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _i} _j}}} + {{{-2}} {{\alpha}} \cdot {{{{ K} _i} _k}} {{{{ K} _l} _j}} {{{{ \gamma} ^k} ^l}}} + {{{4}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ \gamma} _i} _j}}} + {{{-4}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}} \cdot {{{{ \gamma} _i} _j}}}}$
${{{{ \bar{A}} _i} _j} _{,t}} = {{{W}} {{\left({{{{W}} {{{\frac{1}{3}}{\left({{{{3}} {{\left({{{{-1}} {{{{ \alpha} _{,i}} _{,j}}}} + {{{\alpha}} \cdot {{{{ R} _i} _j}}} + {{{{ \beta} ^k}} {{{{{ K} _i} _j} _{,k}}}} + {{{{{ K} _k} _i}} {{{{ \beta} ^k} _{,j}}}} + {{{{{ K} _k} _j}} {{{{ \beta} ^k} _{,i}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,k}}} {{{{ \gamma} ^k} ^m}} {{{{{ \gamma} _i} _j} _{,m}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,k}}} {{{{ \gamma} ^k} ^m}} {{{{{ \gamma} _m} _j} _{,i}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,k}}} {{{{ \gamma} ^k} ^m}} {{{{{ \gamma} _m} _i} _{,j}}}} + {{{K}} {{\alpha}} \cdot {{{{ K} _i} _j}}} + {{{-8}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _i} _j}}} + {{{-2}} {{\alpha}} \cdot {{{{ K} _i} _k}} {{{{ K} _l} _j}} {{{{ \gamma} ^k} ^l}}} + {{{4}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ \gamma} _i} _j}}} + {{{-4}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}} \cdot {{{{ \gamma} _i} _j}}}}\right)}}}{-{{{K}} {{{{{ \gamma} _i} _j} _{,t}}}}}{-{{{{ K} _{,t}}} {{{{ \gamma} _i} _j}}}}}\right)}}}} + {{{2}} {{\left({{{{-1}} \cdot {{\frac{1}{6}}} {{W}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{W}} {{{{ \beta} ^a} _{,a}}}} + {{{{ W} _{,a}}} {{{ \beta} ^a}}} + {{{\frac{1}{3}}} {{K}} {{W}} {{\alpha}}}}\right)}} {{{{ A} _i} _j}}}}\right)}}}$
using ${{ K} _{,t}} = {{{{\alpha}} \cdot {{{K}^{2}}}} + {{{{ K} _{,a}}} {{{ \beta} ^a}}} + {{{{ \alpha} _{,a}}} {{{ \bar{\Lambda}} ^a}} {{{W}^{2}}}} + {{{{ \alpha} _{,a}}} {{{ \hat{\Gamma}} ^a}} {{{W}^{2}}}} + {{{-1}} {{{ \alpha} _{,a}}} {{{ \mathcal{C}} ^a}} {{{W}^{2}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{-1}} {{{{ \alpha} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{W}} {{{ W} _{,a}}} {{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}}} + {{{\alpha}} \cdot {{{{ R} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{4}} {{S}} {{\alpha}} \cdot {{\pi}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^d}} {{{{ \bar{\gamma}} ^c} ^e}} {{{{{ \bar{\gamma}} _d} _e} _{,a}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^d}} {{{{ \bar{\gamma}} ^c} ^e}} {{{{{ \bar{\gamma}} _a} _d} _{,e}}}} + {{{{ \beta} ^a}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^d}} {{{{ \bar{\gamma}} ^c} ^e}} {{{{{ \bar{\gamma}} _a} _e} _{,d}}}} + {{{-12}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}}}}$
${{{{ \bar{A}} _i} _j} _{,t}} = {{{W}} {{\left({{{{W}} {{{\frac{1}{3}}{\left({{{{3}} {{\left({{{{-1}} {{{{ \alpha} _{,i}} _{,j}}}} + {{{\alpha}} \cdot {{{{ R} _i} _j}}} + {{{{ \beta} ^k}} {{{{{ K} _i} _j} _{,k}}}} + {{{{{ K} _k} _i}} {{{{ \beta} ^k} _{,j}}}} + {{{{{ K} _k} _j}} {{{{ \beta} ^k} _{,i}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,k}}} {{{{ \gamma} ^k} ^m}} {{{{{ \gamma} _i} _j} _{,m}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,k}}} {{{{ \gamma} ^k} ^m}} {{{{{ \gamma} _m} _j} _{,i}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,k}}} {{{{ \gamma} ^k} ^m}} {{{{{ \gamma} _m} _i} _{,j}}}} + {{{K}} {{\alpha}} \cdot {{{{ K} _i} _j}}} + {{{-8}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _i} _j}}} + {{{-2}} {{\alpha}} \cdot {{{{ K} _i} _k}} {{{{ K} _l} _j}} {{{{ \gamma} ^k} ^l}}} + {{{4}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ \gamma} _i} _j}}} + {{{-4}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}} \cdot {{{{ \gamma} _i} _j}}}}\right)}}}{-{{{K}} {{{{{ \gamma} _i} _j} _{,t}}}}}{-{{{{{ \gamma} _i} _j}} {{\left({{{{\alpha}} \cdot {{{K}^{2}}}} + {{{{ K} _{,a}}} {{{ \beta} ^a}}} + {{{{ \alpha} _{,a}}} {{{ \bar{\Lambda}} ^a}} {{{W}^{2}}}} + {{{{ \alpha} _{,a}}} {{{ \hat{\Gamma}} ^a}} {{{W}^{2}}}} + {{{-1}} {{{ \alpha} _{,a}}} {{{ \mathcal{C}} ^a}} {{{W}^{2}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{-1}} {{{{ \alpha} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{W}} {{{ W} _{,a}}} {{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}}} + {{{\alpha}} \cdot {{{{ R} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{4}} {{S}} {{\alpha}} \cdot {{\pi}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^d}} {{{{ \bar{\gamma}} ^c} ^e}} {{{{{ \bar{\gamma}} _d} _e} _{,a}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^d}} {{{{ \bar{\gamma}} ^c} ^e}} {{{{{ \bar{\gamma}} _a} _d} _{,e}}}} + {{{{ \beta} ^a}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^d}} {{{{ \bar{\gamma}} ^c} ^e}} {{{{{ \bar{\gamma}} _a} _e} _{,d}}}} + {{{-12}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}}}}\right)}}}}}\right)}}}} + {{{2}} {{\left({{{{-1}} \cdot {{\frac{1}{6}}} {{W}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{W}} {{{{ \beta} ^a} _{,a}}}} + {{{{ W} _{,a}}} {{{ \beta} ^a}}} + {{{\frac{1}{3}}} {{K}} {{W}} {{\alpha}}}}\right)}} {{{{ A} _i} _j}}}}\right)}}}$
simplifying
${{{{ \bar{A}} _i} _j} _{,t}} = {{{{-1}} {{{{ \alpha} _{,i}} _{,j}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{K}} {{{{{ \gamma} _i} _j} _{,t}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{\alpha}} \cdot {{{{ \gamma} _i} _j}} {{{K}^{2}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ A} _i} _j}} {{{W}^{2}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ A} _i} _j}} {{{{ \beta} ^a} _{,a}}} {{{W}^{2}}}} + {{{\alpha}} \cdot {{{{ R} _i} _j}} {{{W}^{2}}}} + {{{{ \beta} ^k}} {{{{{ K} _i} _j} _{,k}}} {{{W}^{2}}}} + {{{{{ K} _k} _j}} {{{{ \beta} ^k} _{,i}}} {{{W}^{2}}}} + {{{{{ K} _k} _i}} {{{{ \beta} ^k} _{,j}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ K} _{,a}}} {{{ \beta} ^a}} {{{{ \gamma} _i} _j}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \alpha} _{,a}}} {{{ \bar{\Lambda}} ^a}} {{{{ \gamma} _i} _j}} {{{W}^{4}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \alpha} _{,a}}} {{{ \hat{\Gamma}} ^a}} {{{{ \gamma} _i} _j}} {{{W}^{4}}}} + {{{\frac{1}{3}}} {{{ \alpha} _{,a}}} {{{ \mathcal{C}} ^a}} {{{{ \gamma} _i} _j}} {{{W}^{4}}}} + {{{\frac{1}{3}}} {{{{ \alpha} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \gamma} _i} _j}} {{{W}^{4}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,k}}} {{{{ \gamma} ^k} ^m}} {{{{{ \gamma} _m} _i} _{,j}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,k}}} {{{{ \gamma} ^k} ^m}} {{{{{ \gamma} _i} _j} _{,m}}} {{{W}^{2}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,k}}} {{{{ \gamma} ^k} ^m}} {{{{{ \gamma} _m} _j} _{,i}}} {{{W}^{2}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{K}} {{\alpha}} \cdot {{{{ A} _i} _j}} {{{W}^{2}}}} + {{{K}} {{\alpha}} \cdot {{{{ K} _i} _j}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{\alpha}} \cdot {{{{ R} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \gamma} _i} _j}} {{{W}^{4}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ W} _{,a}}} {{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \gamma} _i} _j}} {{{W}^{3}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \gamma} _i} _j}} {{{{{ \bar{A}} _b} _c} _{,a}}} {{{W}^{2}}}} + {{{2}} {{W}} {{{ W} _{,a}}} {{{ \beta} ^a}} {{{{ A} _i} _j}}} + {{{-8}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _i} _j}} {{{W}^{2}}}} + {{{8}} \cdot {{\frac{1}{3}}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ \gamma} _i} _j}} {{{W}^{2}}}} + {{{-2}} {{\alpha}} \cdot {{{{ K} _i} _k}} {{{{ K} _l} _j}} {{{{ \gamma} ^k} ^l}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \beta} ^a}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^d}} {{{{ \bar{\gamma}} ^c} ^e}} {{{{ \gamma} _i} _j}} {{{{{ \bar{\gamma}} _a} _e} _{,d}}} {{{W}^{2}}}} + {{{\frac{1}{3}}} {{{ \beta} ^a}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^d}} {{{{ \bar{\gamma}} ^c} ^e}} {{{{ \gamma} _i} _j}} {{{{{ \bar{\gamma}} _d} _e} _{,a}}} {{{W}^{2}}}} + {{{\frac{1}{3}}} {{{ \beta} ^a}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^d}} {{{{ \bar{\gamma}} ^c} ^e}} {{{{ \gamma} _i} _j}} {{{{{ \bar{\gamma}} _a} _d} _{,e}}} {{{W}^{2}}}}}$
${{{{ \bar{A}} _i} _j} _{,t}} = {{{{-1}} {{{{ \alpha} _{,i}} _{,j}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{K}} {{{W}^{2}}} {{\left({{{{{ \beta} ^k}} {{\frac{{{{W}} {{\left({{{{{ \bar{\Gamma}} _i} _k} _j} + {{{{ \bar{\Gamma}} _j} _k} _i}}\right)}}}{-{{{2}} {{{ W} _{,k}}} {{{{ \bar{\gamma}} _i} _j}}}}}{{W}^{3}}}}} + {{{{{ \beta} ^k} _{,i}}} {{\frac{{{ \bar{\gamma}} _j} _k}{{W}^{2}}}}} + {{{{{ \beta} ^k} _{,j}}} {{\frac{{{ \bar{\gamma}} _i} _k}{{W}^{2}}}}}{-{{{2}} {{\alpha}} \cdot {{{\frac{1}{3}}{\left({{{{3}} {{{{{{ \bar{A}} _i} _j}} {{\frac{1}{{W}^{2}}}}}}} + {{{K}} {{\frac{{{ \bar{\gamma}} _i} _j}{{W}^{2}}}}}}\right)}}}}}}\right)}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{\alpha}} \cdot {{\frac{{{ \bar{\gamma}} _i} _j}{{W}^{2}}}} {{{K}^{2}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{{{ \bar{A}} _i} _j}} {{\frac{1}{{W}^{2}}}}}} {{{W}^{2}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{{{ \bar{A}} _i} _j}} {{\frac{1}{{W}^{2}}}}}} {{{{ \beta} ^a} _{,a}}} {{{W}^{2}}}} + {{{\alpha}} \cdot {{{{ R} _i} _j}} {{{W}^{2}}}} + {{{{ \beta} ^k}} {{{W}^{2}}} {{{\left( {\frac{1}{3}}{\left({{{{3}} {{{{{{ \bar{A}} _i} _j}} {{\frac{1}{{W}^{2}}}}}}} + {{{K}} {{\frac{{{ \bar{\gamma}} _i} _j}{{W}^{2}}}}}}\right)}\right)} _{,k}}}} + {{{{\frac{1}{3}}{\left({{{{3}} {{{{{{ \bar{A}} _k} _j}} {{\frac{1}{{W}^{2}}}}}}} + {{{K}} {{\frac{{{ \bar{\gamma}} _k} _j}{{W}^{2}}}}}}\right)}}} {{{{ \beta} ^k} _{,i}}} {{{W}^{2}}}} + {{{{\frac{1}{3}}{\left({{{{3}} {{{{{{ \bar{A}} _k} _i}} {{\frac{1}{{W}^{2}}}}}}} + {{{K}} {{\frac{{{ \bar{\gamma}} _k} _i}{{W}^{2}}}}}}\right)}}} {{{{ \beta} ^k} _{,j}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ K} _{,a}}} {{{ \beta} ^a}} {{\frac{{{ \bar{\gamma}} _i} _j}{{W}^{2}}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \alpha} _{,a}}} {{{ \bar{\Lambda}} ^a}} {{\frac{{{ \bar{\gamma}} _i} _j}{{W}^{2}}}} {{{W}^{4}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \alpha} _{,a}}} {{{ \hat{\Gamma}} ^a}} {{\frac{{{ \bar{\gamma}} _i} _j}{{W}^{2}}}} {{{W}^{4}}}} + {{{\frac{1}{3}}} {{{ \alpha} _{,a}}} {{{ \mathcal{C}} ^a}} {{\frac{{{ \bar{\gamma}} _i} _j}{{W}^{2}}}} {{{W}^{4}}}} + {{{\frac{1}{3}}} {{{{ \alpha} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}} {{\frac{{{ \bar{\gamma}} _i} _j}{{W}^{2}}}} {{{W}^{4}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,k}}} {{{{{{ \bar{\gamma}} ^k} ^m}} {{{W}^{2}}}}} {{\frac{{{{W}} {{\left({{{{{ \bar{\Gamma}} _m} _j} _i} + {{{{ \bar{\Gamma}} _i} _j} _m}}\right)}}}{-{{{2}} {{{ W} _{,j}}} {{{{ \bar{\gamma}} _m} _i}}}}}{{W}^{3}}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,k}}} {{{{{{ \bar{\gamma}} ^k} ^m}} {{{W}^{2}}}}} {{\frac{{{{W}} {{\left({{{{{ \bar{\Gamma}} _i} _m} _j} + {{{{ \bar{\Gamma}} _j} _m} _i}}\right)}}}{-{{{2}} {{{ W} _{,m}}} {{{{ \bar{\gamma}} _i} _j}}}}}{{W}^{3}}}} {{{W}^{2}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,k}}} {{{{{{ \bar{\gamma}} ^k} ^m}} {{{W}^{2}}}}} {{\frac{{{{W}} {{\left({{{{{ \bar{\Gamma}} _m} _i} _j} + {{{{ \bar{\Gamma}} _j} _i} _m}}\right)}}}{-{{{2}} {{{ W} _{,i}}} {{{{ \bar{\gamma}} _m} _j}}}}}{{W}^{3}}}} {{{W}^{2}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{K}} {{\alpha}} \cdot {{{{{{ \bar{A}} _i} _j}} {{\frac{1}{{W}^{2}}}}}} {{{W}^{2}}}} + {{{K}} {{\alpha}} \cdot {{{\frac{1}{3}}{\left({{{{3}} {{{{{{ \bar{A}} _i} _j}} {{\frac{1}{{W}^{2}}}}}}} + {{{K}} {{\frac{{{ \bar{\gamma}} _i} _j}{{W}^{2}}}}}}\right)}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{\alpha}} \cdot {{{{ R} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{\frac{{{ \bar{\gamma}} _i} _j}{{W}^{2}}}} {{{W}^{4}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ W} _{,a}}} {{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}} {{\frac{{{ \bar{\gamma}} _i} _j}{{W}^{2}}}} {{{W}^{3}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{\frac{{{ \bar{\gamma}} _i} _j}{{W}^{2}}}} {{{{{ \bar{A}} _b} _c} _{,a}}} {{{W}^{2}}}} + {{{2}} {{W}} {{{ W} _{,a}}} {{{ \beta} ^a}} {{{{{{ \bar{A}} _i} _j}} {{\frac{1}{{W}^{2}}}}}}} + {{{-8}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _i} _j}} {{{W}^{2}}}} + {{{8}} \cdot {{\frac{1}{3}}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{\frac{{{ \bar{\gamma}} _i} _j}{{W}^{2}}}} {{{W}^{2}}}} + {{{-2}} {{\alpha}} \cdot {{{\frac{1}{3}}{\left({{{{3}} {{{{{{ \bar{A}} _i} _k}} {{\frac{1}{{W}^{2}}}}}}} + {{{K}} {{\frac{{{ \bar{\gamma}} _i} _k}{{W}^{2}}}}}}\right)}}} {{{\frac{1}{3}}{\left({{{{3}} {{{{{{ \bar{A}} _l} _j}} {{\frac{1}{{W}^{2}}}}}}} + {{{K}} {{\frac{{{ \bar{\gamma}} _l} _j}{{W}^{2}}}}}}\right)}}} {{{{{{ \bar{\gamma}} ^k} ^l}} {{{W}^{2}}}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \beta} ^a}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^d}} {{{{ \bar{\gamma}} ^c} ^e}} {{\frac{{{ \bar{\gamma}} _i} _j}{{W}^{2}}}} {{\left({{{{{ \bar{\Gamma}} _a} _d} _e} + {{{{ \bar{\Gamma}} _e} _d} _a}}\right)}} {{{W}^{2}}}} + {{{\frac{1}{3}}} {{{ \beta} ^a}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^d}} {{{{ \bar{\gamma}} ^c} ^e}} {{\frac{{{ \bar{\gamma}} _i} _j}{{W}^{2}}}} {{\left({{{{{ \bar{\Gamma}} _d} _a} _e} + {{{{ \bar{\Gamma}} _e} _a} _d}}\right)}} {{{W}^{2}}}} + {{{\frac{1}{3}}} {{{ \beta} ^a}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^d}} {{{{ \bar{\gamma}} ^c} ^e}} {{\frac{{{ \bar{\gamma}} _i} _j}{{W}^{2}}}} {{\left({{{{{ \bar{\Gamma}} _a} _e} _d} + {{{{ \bar{\Gamma}} _d} _e} _a}}\right)}} {{{W}^{2}}}}}$
${{{{ \bar{A}} _i} _j} _{,t}} = {{{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{A}} _i} _j}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{A}} _i} _j}} {{{{ \beta} ^a} _{,a}}}} + {{{{ \beta} ^a}} {{{{{ \bar{A}} _i} _j} _{,a}}}} + {{{-1}} {{{{ \alpha} _{,i}} _{,j}}} {{{W}^{2}}}} + {{{{{ \bar{A}} _a} _i}} {{{{ \beta} ^a} _{,j}}}} + {{{{{ \bar{A}} _a} _j}} {{{{ \beta} ^a} _{,i}}}} + {{{K}} {{\alpha}} \cdot {{{{ \bar{A}} _i} _j}}} + {{{-1}} {{W}} {{{ W} _{,i}}} {{{ \alpha} _{,j}}}} + {{{-1}} {{W}} {{{ W} _{,j}}} {{{ \alpha} _{,i}}}} + {{{\alpha}} \cdot {{{{ R} _i} _j}} {{{W}^{2}}}} + {{{{ \alpha} _{,a}}} {{{{{ \bar{\Gamma}} ^a} _i} _j}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \alpha} _{,a}}} {{{ \bar{\Lambda}} ^a}} {{{{ \bar{\gamma}} _i} _j}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \alpha} _{,a}}} {{{ \hat{\Gamma}} ^a}} {{{{ \bar{\gamma}} _i} _j}} {{{W}^{2}}}} + {{{\frac{1}{3}}} {{{ \alpha} _{,a}}} {{{ \mathcal{C}} ^a}} {{{{ \bar{\gamma}} _i} _j}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} _i} _j}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{\frac{1}{3}}} {{{{ \alpha} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} _i} _j}} {{{W}^{2}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{ \beta} ^a}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} _i} _j}} {{{{{ \bar{\Gamma}} ^b} ^c} _a}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{W}} {{{ W} _{,a}}} {{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} _i} _j}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{\alpha}} \cdot {{{{ R} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} _i} _j}} {{{W}^{2}}}} + {{{8}} \cdot {{\frac{1}{3}}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ \bar{\gamma}} _i} _j}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _b} _j}} {{{{ \bar{A}} _i} _a}} {{{{ \bar{\gamma}} ^a} ^b}}} + {{{W}} {{{ W} _{,a}}} {{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^b} ^a}} {{{{ \bar{\gamma}} _i} _j}}} + {{{-8}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _i} _j}} {{{W}^{2}}}}}$
${{{{ \bar{A}} _i} _j} _{,t}} = {{{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{A}} _i} _j}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{A}} _i} _j}} {{{{ \beta} ^a} _{,a}}}} + {{{{ \beta} ^a}} {{{{{ \bar{A}} _i} _j} _{,a}}}} + {{{-1}} {{{{ \alpha} _{,i}} _{,j}}} {{{W}^{2}}}} + {{{{{ \bar{A}} _a} _i}} {{{{ \beta} ^a} _{,j}}}} + {{{{{ \bar{A}} _a} _j}} {{{{ \beta} ^a} _{,i}}}} + {{{K}} {{\alpha}} \cdot {{{{ \bar{A}} _i} _j}}} + {{{-1}} {{W}} {{{ W} _{,j}}} {{{ \alpha} _{,i}}}} + {{{-1}} {{W}} {{{ W} _{,i}}} {{{ \alpha} _{,j}}}} + {{{\alpha}} \cdot {{{{ R} _i} _j}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \alpha} _{,a}}} {{{ \bar{\Lambda}} ^a}} {{{{ \bar{\gamma}} _i} _j}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \alpha} _{,a}}} {{{ \hat{\Gamma}} ^a}} {{{{ \bar{\gamma}} _i} _j}} {{{W}^{2}}}} + {{{\frac{1}{3}}} {{{ \alpha} _{,a}}} {{{ \mathcal{C}} ^a}} {{{{ \bar{\gamma}} _i} _j}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} _i} _j}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{\frac{1}{3}}} {{{{ \alpha} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} _i} _j}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{{ \bar{\gamma}} _i} _j} _{,b}}} {{{W}^{2}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{{ \bar{\gamma}} _b} _i} _{,j}}} {{{W}^{2}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{{ \bar{\gamma}} _b} _j} _{,i}}} {{{W}^{2}}}} + {{{8}} \cdot {{\frac{1}{3}}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ \bar{\gamma}} _i} _j}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{\alpha}} \cdot {{{{ R} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} _i} _j}} {{{W}^{2}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _a} _j}} {{{{ \bar{A}} _b} _i}} {{{{ \bar{\gamma}} ^a} ^b}}} + {{{2}} \cdot {{\frac{1}{3}}} {{W}} {{{ W} _{,a}}} {{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} _i} _j}}} + {{{-8}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _i} _j}} {{{W}^{2}}}} + {{{\frac{1}{3}}} {{{ \beta} ^a}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^d}} {{{{ \bar{\gamma}} ^c} ^e}} {{{{ \bar{\gamma}} _i} _j}} {{{{{ \bar{\gamma}} _d} _e} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \beta} ^a}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^d}} {{{{ \bar{\gamma}} ^c} ^e}} {{{{ \bar{\gamma}} _i} _j}} {{{{{ \bar{\gamma}} _a} _e} _{,d}}}} + {{{\frac{1}{3}}} {{{ \beta} ^a}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^d}} {{{{ \bar{\gamma}} ^c} ^e}} {{{{ \bar{\gamma}} _i} _j}} {{{{{ \bar{\gamma}} _a} _d} _{,e}}}}}$

using locally-Minkowski normalized coordinates:
${{{{{ e} _i} ^I}} {{{{ e} _j} ^J}} {{{{{ \bar{A}} _I} _J} _{,t}}}} = {{{{-1}} {{{{ \alpha} _{,i}} _{,j}}} {{{W}^{2}}}} + {{{-1}} {{W}} {{{ W} _{,j}}} {{{ \alpha} _{,i}}}} + {{{-1}} {{W}} {{{ W} _{,i}}} {{{ \alpha} _{,j}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^A}} {{{{ \bar{A}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _I} _J}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}} {{{{{ e} ^a} _A} _{,a}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{A}} _I} _J}} {{{{ \beta} ^A} _{,a}}} {{{{ e} ^a} _A}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}}} + {{{K}} {{\alpha}} \cdot {{{{ \bar{A}} _I} _J}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}}} + {{{\alpha}} \cdot {{{{ R} _I} _J}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}} {{{W}^{2}}}} + {{{{ \beta} ^A}} {{{{ \bar{A}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} _j} ^J}} {{{{{ e} _i} ^I} _{,a}}}} + {{{{ \beta} ^A}} {{{{ e} ^a} _A}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}} {{{{{ \bar{A}} _I} _J} _{,a}}}} + {{{{ \beta} ^B}} {{{{ \bar{A}} _A} _J}} {{{{ e} _a} ^A}} {{{{ e} _j} ^J}} {{{{{ e} ^a} _B} _{,i}}}} + {{{{ \beta} ^A}} {{{{ \bar{A}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} _i} ^I}} {{{{{ e} _j} ^J} _{,a}}}} + {{{{ \beta} ^B}} {{{{ \bar{A}} _A} _I}} {{{{ e} _a} ^A}} {{{{ e} _i} ^I}} {{{{{ e} ^a} _B} _{,j}}}} + {{{{{ \bar{A}} _A} _I}} {{{{ \beta} ^B} _{,j}}} {{{{ e} _a} ^A}} {{{{ e} ^a} _B}} {{{{ e} _i} ^I}}} + {{{{{ \bar{A}} _A} _J}} {{{{ \beta} ^B} _{,i}}} {{{{ e} _a} ^A}} {{{{ e} ^a} _B}} {{{{ e} _j} ^J}}} + {{{8}} \cdot {{\frac{1}{3}}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ \bar{\gamma}} _I} _J}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{\alpha}} \cdot {{{{ R} _A} _B}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \alpha} _{,a}}} {{{ \bar{\Lambda}} ^A}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \alpha} _{,a}}} {{{ \hat{\Gamma}} ^A}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}} {{{W}^{2}}}} + {{{\frac{1}{3}}} {{{ \alpha} _{,a}}} {{{ \mathcal{C}} ^A}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}} {{{W}^{2}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _A} _J}} {{{{ \bar{A}} _B} _I}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}}} + {{{-8}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _I} _J}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}} {{{W}^{2}}}} + {{{\frac{1}{3}}} {{{{ \alpha} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}} {{{W}^{2}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} _C} _I}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _i} ^I}} {{{{{ e} _b} ^C} _{,j}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _i} ^I}} {{{{{ e} _j} ^J} _{,b}}} {{{W}^{2}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} _C} _I}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^C}} {{{{{ e} _i} ^I} _{,j}}} {{{W}^{2}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} _C} _J}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^C}} {{{{{ e} _j} ^J} _{,i}}} {{{W}^{2}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} _C} _J}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _j} ^J}} {{{{{ e} _b} ^C} _{,i}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _j} ^J}} {{{{{ e} _i} ^I} _{,b}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}} {{{{{ \bar{\gamma}} _I} _J} _{,b}}} {{{W}^{2}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^C}} {{{{ e} _j} ^J}} {{{{{ \bar{\gamma}} _C} _J} _{,i}}} {{{W}^{2}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^C}} {{{{ e} _i} ^I}} {{{{{ \bar{\gamma}} _C} _I} _{,j}}} {{{W}^{2}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{W}} {{{ W} _{,a}}} {{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _D} _E}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^D}} {{{{ e} ^c} _C}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}} {{{{{ e} _c} ^E} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _D} _E}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _c} ^E}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}} {{{{{ e} _b} ^D} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _b} ^D}} {{{{ e} ^c} _C}} {{{{ e} _c} ^E}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}} {{{{{ \bar{A}} _D} _E} _{,a}}}} + {{{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} ^d} _D}} {{{{ e} _d} ^F}} {{{{ e} ^e} _E}} {{{{ e} _e} ^G}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}} {{{{{ \bar{\gamma}} _F} _G} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} _a} ^F}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{ e} _e} ^G}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}} {{{{{ \bar{\gamma}} _F} _G} _{,d}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _F} _G}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} _a} ^F}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}} {{{{{ e} _e} ^G} _{,d}}}} + {{{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _F} _G}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} _a} ^F}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}} {{{{{ e} _d} ^G} _{,e}}}} + {{{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} _a} ^F}} {{{{ e} ^d} _D}} {{{{ e} _d} ^G}} {{{{ e} ^e} _E}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}} {{{{{ \bar{\gamma}} _F} _G} _{,e}}}} + {{{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _F} _G}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} ^d} _D}} {{{{ e} _d} ^F}} {{{{ e} ^e} _E}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}} {{{{{ e} _e} ^G} _{,a}}}} + {{{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _F} _G}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{ e} _e} ^G}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}} {{{{{ e} _d} ^F} _{,a}}}} + {{{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _F} _G}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} ^d} _D}} {{{{ e} _d} ^G}} {{{{ e} ^e} _E}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}} {{{{{ e} _a} ^F} _{,e}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _F} _G}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{ e} _e} ^G}} {{{{ e} _i} ^I}} {{{{ e} _j} ^J}} {{{{{ e} _a} ^F} _{,d}}}}}$
cancelling forward and inverse transforms, and simplifying deltas...
${{{{ \bar{A}} _I} _J} _{,t}} = {{{{-1}} {{{{ \alpha} _{,i}} _{,j}}} {{{{ e} ^i} _I}} {{{{ e} ^j} _J}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^A}} {{{{ \bar{A}} _I} _J}} {{{{ e} ^a} _A}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _I} _J}} {{{{{ e} ^a} _A} _{,a}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{A}} _I} _J}} {{{{ \beta} ^A} _{,a}}} {{{{ e} ^a} _A}}} + {{{K}} {{\alpha}} \cdot {{{{ \bar{A}} _I} _J}}} + {{{-1}} {{W}} {{{ W} _{,i}}} {{{ \alpha} _{,j}}} {{{{ e} ^i} _I}} {{{{ e} ^j} _J}}} + {{{-1}} {{W}} {{{ W} _{,j}}} {{{ \alpha} _{,i}}} {{{{ e} ^i} _I}} {{{{ e} ^j} _J}}} + {{{\alpha}} \cdot {{{{ R} _I} _J}} {{{W}^{2}}}} + {{{{ \beta} ^A}} {{{{ e} ^a} _A}} {{{{{ \bar{A}} _I} _J} _{,a}}}} + {{{{{ \bar{A}} _A} _J}} {{{{ \beta} ^A} _{,i}}} {{{{ e} ^i} _I}}} + {{{{{ \bar{A}} _A} _I}} {{{{ \beta} ^A} _{,j}}} {{{{ e} ^j} _J}}} + {{{8}} \cdot {{\frac{1}{3}}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ \bar{\gamma}} _I} _J}}} + {{{{ \beta} ^A}} {{{{ \bar{A}} _I} _N}} {{{{ e} ^a} _A}} {{{{ e} ^j} _J}} {{{{{ e} _j} ^N} _{,a}}}} + {{{{ \beta} ^A}} {{{{ \bar{A}} _M} _J}} {{{{ e} ^a} _A}} {{{{ e} ^i} _I}} {{{{{ e} _i} ^M} _{,a}}}} + {{{{ \beta} ^B}} {{{{ \bar{A}} _A} _J}} {{{{ e} _a} ^A}} {{{{ e} ^i} _I}} {{{{{ e} ^a} _B} _{,i}}}} + {{{{ \beta} ^B}} {{{{ \bar{A}} _A} _I}} {{{{ e} _a} ^A}} {{{{ e} ^j} _J}} {{{{{ e} ^a} _B} _{,j}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{\alpha}} \cdot {{{{ R} _A} _B}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} _I} _J}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \alpha} _{,a}}} {{{ \bar{\Lambda}} ^A}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \alpha} _{,a}}} {{{ \hat{\Gamma}} ^A}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{W}^{2}}}} + {{{\frac{1}{3}}} {{{ \alpha} _{,a}}} {{{ \mathcal{C}} ^A}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{W}^{2}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _A} _J}} {{{{ \bar{A}} _B} _I}} {{{{ \bar{\gamma}} ^A} ^B}}} + {{{-8}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _I} _J}} {{{W}^{2}}}} + {{{\frac{1}{3}}} {{{{ \alpha} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{{ \bar{\gamma}} _I} _J} _{,b}}} {{{W}^{2}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ e} ^a} _A}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _B} _J} _{,i}}} {{{W}^{2}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ e} ^a} _A}} {{{{ e} ^j} _J}} {{{{{ \bar{\gamma}} _B} _I} _{,j}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{{ \bar{A}} _B} _C} _{,a}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ e} ^a} _M}} {{{{ e} ^i} _I}} {{{{ e} ^j} _J}} {{{{{ e} _i} ^M} _{,j}}} {{{W}^{2}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} _C} _I}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^j} _J}} {{{{{ e} _b} ^C} _{,j}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} _M} _J}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^i} _I}} {{{{{ e} _i} ^M} _{,b}}} {{{W}^{2}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} _C} _J}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^i} _I}} {{{{{ e} _b} ^C} _{,i}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} _I} _N}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^j} _J}} {{{{{ e} _j} ^N} _{,b}}} {{{W}^{2}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ e} ^a} _N}} {{{{ e} ^i} _I}} {{{{ e} ^j} _J}} {{{{{ e} _j} ^N} _{,i}}} {{{W}^{2}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{W}} {{{ W} _{,a}}} {{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _D} _C}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{{ e} _b} ^D} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _E}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{{ e} _c} ^E} _{,a}}}} + {{{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^e} _E}} {{{{{ \bar{\gamma}} _A} _D} _{,e}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^d} _D}} {{{{{ \bar{\gamma}} _A} _E} _{,d}}}} + {{{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{{ \bar{\gamma}} _D} _E} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _A} _G}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{{ e} _e} ^G} _{,d}}}} + {{{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} ^e} _E}} {{{{{ e} _a} ^B} _{,e}}}} + {{{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} ^d} _D}} {{{{{ e} _d} ^C} _{,a}}}} + {{{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _A} _G}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{{ e} _d} ^G} _{,e}}}} + {{{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _G} _C}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} ^e} _E}} {{{{{ e} _e} ^G} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} ^d} _D}} {{{{{ e} _a} ^C} _{,d}}}}}$

---

collecting partial derivatives:
${{ \alpha} _{,t}} = {{{{{ \alpha} _{,i}}} {{{ \beta} ^i}}}{-{{{{\alpha}^{2}}} {{f}} {{K}}}}}$
${{{ \beta} ^i} _{,t}} = {{ B} ^i}$
${{{ B} ^i} _{,t}} = {{{{\frac{3}{4}}} {{{{ \bar{\Lambda}} ^i} _{,t}}}}{-{{{\eta}} \cdot {{{ B} ^i}}}}}$
${{ W} _{,t}} = {{{{-1}} \cdot {{\frac{1}{6}}} {{W}} {{{ \bar{\gamma}} _{,i}}} {{{ \beta} ^i}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{W}} {{{{ \beta} ^i} _{,i}}}} + {{{{ W} _{,i}}} {{{ \beta} ^i}}} + {{{\frac{1}{3}}} {{K}} {{W}} {{\alpha}}}}$
${{ K} _{,t}} = {{{{\alpha}} \cdot {{{K}^{2}}}} + {{{{ K} _{,a}}} {{{ \beta} ^a}}} + {{{{ \alpha} _{,a}}} {{{ \bar{\Lambda}} ^a}} {{{W}^{2}}}} + {{{{ \alpha} _{,a}}} {{{ \hat{\Gamma}} ^a}} {{{W}^{2}}}} + {{{-1}} {{{ \alpha} _{,a}}} {{{ \mathcal{C}} ^a}} {{{W}^{2}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{-1}} {{{{ \alpha} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{W}} {{{ W} _{,a}}} {{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}}} + {{{\alpha}} \cdot {{{{ R} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{W}^{2}}}} + {{{4}} {{S}} {{\alpha}} \cdot {{\pi}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^d}} {{{{ \bar{\gamma}} ^c} ^e}} {{{{{ \bar{\gamma}} _d} _e} _{,a}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^d}} {{{{ \bar{\gamma}} ^c} ^e}} {{{{{ \bar{\gamma}} _a} _d} _{,e}}}} + {{{{ \beta} ^a}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^d}} {{{{ \bar{\gamma}} ^c} ^e}} {{{{{ \bar{\gamma}} _a} _e} _{,d}}}} + {{{-12}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}}}}$
${{{{ \bar{\epsilon}} _i} _j} _{,t}} = {{{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,k}}} {{{ \beta} ^k}} {{{{ \bar{\gamma}} _i} _j}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} _i} _j}} {{{{ \beta} ^k} _{,k}}}} + {{{{ \beta} ^k}} {{{{{ \bar{\gamma}} _i} _j} _{,k}}}} + {{{{{ \bar{\gamma}} _j} _k}} {{{{ \beta} ^k} _{,i}}}} + {{{{{ \bar{\gamma}} _i} _k}} {{{{ \beta} ^k} _{,j}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _i} _j}}}}$
${{{{ \bar{A}} _i} _j} _{,t}} = {{{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{A}} _i} _j}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{A}} _i} _j}} {{{{ \beta} ^a} _{,a}}}} + {{{{ \beta} ^a}} {{{{{ \bar{A}} _i} _j} _{,a}}}} + {{{-1}} {{{{ \alpha} _{,i}} _{,j}}} {{{W}^{2}}}} + {{{{{ \bar{A}} _a} _i}} {{{{ \beta} ^a} _{,j}}}} + {{{{{ \bar{A}} _a} _j}} {{{{ \beta} ^a} _{,i}}}} + {{{K}} {{\alpha}} \cdot {{{{ \bar{A}} _i} _j}}} + {{{-1}} {{W}} {{{ W} _{,j}}} {{{ \alpha} _{,i}}}} + {{{-1}} {{W}} {{{ W} _{,i}}} {{{ \alpha} _{,j}}}} + {{{\alpha}} \cdot {{{{ R} _i} _j}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \alpha} _{,a}}} {{{ \bar{\Lambda}} ^a}} {{{{ \bar{\gamma}} _i} _j}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \alpha} _{,a}}} {{{ \hat{\Gamma}} ^a}} {{{{ \bar{\gamma}} _i} _j}} {{{W}^{2}}}} + {{{\frac{1}{3}}} {{{ \alpha} _{,a}}} {{{ \mathcal{C}} ^a}} {{{{ \bar{\gamma}} _i} _j}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} _i} _j}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{\frac{1}{3}}} {{{{ \alpha} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} _i} _j}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{{ \bar{\gamma}} _i} _j} _{,b}}} {{{W}^{2}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{{ \bar{\gamma}} _b} _i} _{,j}}} {{{W}^{2}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{{ \bar{\gamma}} _b} _j} _{,i}}} {{{W}^{2}}}} + {{{8}} \cdot {{\frac{1}{3}}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ \bar{\gamma}} _i} _j}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{\alpha}} \cdot {{{{ R} _a} _b}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} _i} _j}} {{{W}^{2}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _a} _j}} {{{{ \bar{A}} _b} _i}} {{{{ \bar{\gamma}} ^a} ^b}}} + {{{2}} \cdot {{\frac{1}{3}}} {{W}} {{{ W} _{,a}}} {{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} _i} _j}}} + {{{-8}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _i} _j}} {{{W}^{2}}}} + {{{\frac{1}{3}}} {{{ \beta} ^a}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^d}} {{{{ \bar{\gamma}} ^c} ^e}} {{{{ \bar{\gamma}} _i} _j}} {{{{{ \bar{\gamma}} _d} _e} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \beta} ^a}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^d}} {{{{ \bar{\gamma}} ^c} ^e}} {{{{ \bar{\gamma}} _i} _j}} {{{{{ \bar{\gamma}} _a} _e} _{,d}}}} + {{{\frac{1}{3}}} {{{ \beta} ^a}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^b} ^d}} {{{{ \bar{\gamma}} ^c} ^e}} {{{{ \bar{\gamma}} _i} _j}} {{{{{ \bar{\gamma}} _a} _d} _{,e}}}}}$
${{{ \bar{\Lambda}} ^i} _{,t}} = {{{{\frac{1}{6}}} {{{ \beta} ^i}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{{ \beta} ^b} _{,a}} _{,b}}}} + {{{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \beta} ^a} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \bar{\gamma}} _{,b}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{\frac{1}{{\bar{\gamma}}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \bar{\gamma}} _{,b}}} {{{ \beta} ^b}} {{{{ \bar{\gamma}} ^a} ^i}} {{\frac{1}{{\bar{\gamma}}^{2}}}}} + {{{{{ \bar{\gamma}} ^a} ^b}} {{{{{ \beta} ^i} _{,a}} _{,b}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \beta} ^b} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{2}}} {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{\gamma}} ^b} ^a}} {{{{ \beta} ^i} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \beta} ^c} _{,c}}} {{{{{ \hat{\Gamma}} ^i} _a} _b}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \hat{\Gamma}} ^i} _b} _c}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{{ \bar{\gamma}} _c} _d} _{,a}} _{,b}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _b} _d} _{,a}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,b}}} {{{{{ \bar{\gamma}} _c} _d} _{,a}}}} + {{{2}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \beta} ^c} _{,a}}} {{{{{ \hat{\Gamma}} ^i} _b} _c}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{ \beta} ^d} _{,d}}} {{{{{ \bar{\gamma}} _a} _c} _{,b}}}} + {{{\alpha}} \cdot {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _b} _c} _{,a}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{{{ \bar{\gamma}} _b} _c} _{,a}} _{,d}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,b}}} {{{{{ \bar{\gamma}} _a} _c} _{,d}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,a}}} {{{{{ \bar{\gamma}} _b} _d} _{,c}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \beta} ^i} _{,c}}} {{{{{ \bar{\gamma}} _a} _d} _{,b}}}} + {{{-1}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^c} ^i}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _a} _d} _{,b}}}} + {{{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \beta} ^d} _{,c}}} {{{{{ \bar{\gamma}} _a} _b} _{,d}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^i}} {{{{{ \bar{\gamma}} _b} _d} _{,c}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{2}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^b}} {{{{ \bar{\gamma}} ^c} ^a}} {{{{ \bar{\gamma}} ^d} ^i}} {{{{{ \bar{\gamma}} _c} _d} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{\gamma}} ^a} ^i}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{{ \bar{A}} _a} _b} _{,c}}}} + {{{-2}} {{{ \alpha} _{,a}}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^a} ^c}} {{{{ \bar{\gamma}} ^i} ^b}}} + {{{-1}} {{\alpha}} \cdot {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{A}} _b} _c}} {{{{ \bar{\gamma}} ^a} ^b}} {{{{ \bar{\gamma}} ^i} ^c}} {{\frac{1}{\bar{\gamma}}}}} + {{{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{{ \bar{\gamma}} _b} _d} _{,a}}} {{{{{ \hat{\Gamma}} ^i} _c} _e}}} + {{{\frac{1}{2}}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _c} _e} _{,a}}} {{{{{ \bar{\gamma}} _d} _f} _{,b}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _a} _b}} {{{{ \bar{\gamma}} ^c} ^a}} {{{{ \bar{\gamma}} ^d} ^b}} {{{{{ \hat{\Gamma}} ^i} _c} _d}}} + {{{-1}} {{\alpha}} \cdot {{{{ \bar{A}} _a} _b}} {{{{ \bar{\gamma}} ^c} ^a}} {{{{ \bar{\gamma}} ^d} ^b}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{{ \bar{\gamma}} _c} _d} _{,e}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^i}} {{{{ \bar{\gamma}} ^c} ^d}} {{{{ \bar{\gamma}} ^e} ^f}} {{{{{ \bar{\gamma}} _b} _d} _{,f}}} {{{{{ \bar{\gamma}} _c} _e} _{,a}}}} + {{{-1}} {{{ \beta} ^a}} {{{{ \bar{\gamma}} ^b} ^c}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} ^f} ^i}} {{{{{ \bar{\gamma}} _b} _e} _{,c}}} {{{{{ \bar{\gamma}} _d} _f} _{,a}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _a} _b}} {{{{ \bar{\gamma}} ^c} ^a}} {{{{ \bar{\gamma}} ^d} ^e}} {{{{ \bar{\gamma}} ^i} ^b}} {{{{{ \bar{\gamma}} _c} _d} _{,e}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _a} _b}} {{{{ \bar{\gamma}} ^c} ^a}} {{{{ \bar{\gamma}} ^d} ^b}} {{{{ \bar{\gamma}} ^e} ^i}} {{{{{ \bar{\gamma}} _c} _e} _{,d}}}}}$

locally-Minkowski:
${{ \alpha} _{,t}} = {{{{{ \alpha} _{,i}}} {{{ \beta} ^I}} {{{{ e} ^i} _I}}} + {{{-1}} {{K}} {{f}} {{{\alpha}^{2}}}}}$
${{{ \beta} ^I} _{,t}} = {{ B} ^I}$
${{{ B} ^I} _{,t}} = {{{{3}} \cdot {{\frac{1}{4}}} {{{{ \bar{\Lambda}} ^I} _{,t}}}} + {{{-1}} {{\eta}} \cdot {{{ B} ^I}}}}$
${{ W} _{,t}} = {{{{-1}} \cdot {{\frac{1}{6}}} {{W}} {{{ \bar{\gamma}} _{,i}}} {{{ \beta} ^I}} {{{{ e} ^i} _I}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{3}}} {{K}} {{W}} {{\alpha}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{W}} {{{ \beta} ^I}} {{{{{ e} ^i} _I} _{,i}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{W}} {{{{ \beta} ^I} _{,i}}} {{{{ e} ^i} _I}}} + {{{{ W} _{,i}}} {{{ \beta} ^I}} {{{{ e} ^i} _I}}}}$
${{ K} _{,t}} = {{{{\alpha}} \cdot {{{K}^{2}}}} + {{{{ K} _{,a}}} {{{ \beta} ^A}} {{{{ e} ^a} _A}}} + {{{\alpha}} \cdot {{{{ R} _A} _B}} {{{{ \bar{\gamma}} ^A} ^B}} {{{W}^{2}}}} + {{{{ \alpha} _{,a}}} {{{ \bar{\Lambda}} ^A}} {{{{ e} ^a} _A}} {{{W}^{2}}}} + {{{{ \alpha} _{,a}}} {{{ \hat{\Gamma}} ^A}} {{{{ e} ^a} _A}} {{{W}^{2}}}} + {{{-1}} {{{ \alpha} _{,a}}} {{{ \mathcal{C}} ^A}} {{{{ e} ^a} _A}} {{{W}^{2}}}} + {{{4}} {{S}} {{\alpha}} \cdot {{\pi}}} + {{{-1}} {{{{ \alpha} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{W}^{2}}}} + {{{-12}} {{\alpha}} \cdot {{\pi}} \cdot {{\rho}}} + {{{W}} {{{ W} _{,a}}} {{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}}} + {{{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{{ \bar{A}} _B} _C} _{,a}}}} + {{{{ \beta} ^A}} {{{{ \bar{A}} _D} _C}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{{ e} _b} ^D} _{,a}}}} + {{{{ \beta} ^A}} {{{{ \bar{A}} _B} _E}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{{ e} _c} ^E} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ e} ^a} _A}} {{{{{ \bar{\gamma}} _D} _E} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ e} ^e} _E}} {{{{{ \bar{\gamma}} _A} _D} _{,e}}}} + {{{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ e} ^d} _D}} {{{{{ \bar{\gamma}} _A} _E} _{,d}}}} + {{{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ e} ^a} _A}} {{{{ e} ^d} _D}} {{{{{ e} _a} ^C} _{,d}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _A} _G}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{{ e} _d} ^G} _{,e}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ e} ^a} _A}} {{{{ e} ^d} _D}} {{{{{ e} _d} ^C} _{,a}}}} + {{{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _A} _G}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{{ e} _e} ^G} _{,d}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{A}} _G} _C}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ e} ^a} _A}} {{{{ e} ^e} _E}} {{{{{ e} _e} ^G} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ e} ^a} _A}} {{{{ e} ^e} _E}} {{{{{ e} _a} ^B} _{,e}}}}}$
${{{{ \bar{\epsilon}} _I} _J} _{,t}} = {{{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,k}}} {{{ \beta} ^K}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^k} _K}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _I} _J}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{ \beta} ^K}} {{{{ \bar{\gamma}} _I} _J}} {{{{{ e} ^k} _K} _{,k}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} _I} _J}} {{{{ \beta} ^K} _{,k}}} {{{{ e} ^k} _K}}} + {{{{ \beta} ^K}} {{{{ e} ^k} _K}} {{{{{ \bar{\gamma}} _I} _J} _{,k}}}} + {{{{{ \bar{\gamma}} _J} _A}} {{{{ \beta} ^A} _{,i}}} {{{{ e} ^i} _I}}} + {{{{{ \bar{\gamma}} _I} _A}} {{{{ \beta} ^A} _{,j}}} {{{{ e} ^j} _J}}} + {{{{ \beta} ^K}} {{{{ \bar{\gamma}} _M} _J}} {{{{ e} ^i} _I}} {{{{ e} ^k} _K}} {{{{{ e} _i} ^M} _{,k}}}} + {{{{ \beta} ^A}} {{{{ \bar{\gamma}} _J} _K}} {{{{ e} ^i} _I}} {{{{ e} _k} ^K}} {{{{{ e} ^k} _A} _{,i}}}} + {{{{ \beta} ^K}} {{{{ \bar{\gamma}} _I} _N}} {{{{ e} ^j} _J}} {{{{ e} ^k} _K}} {{{{{ e} _j} ^N} _{,k}}}} + {{{{ \beta} ^A}} {{{{ \bar{\gamma}} _I} _K}} {{{{ e} ^j} _J}} {{{{ e} _k} ^K}} {{{{{ e} ^k} _A} _{,j}}}}}$
${{{{ \bar{A}} _I} _J} _{,t}} = {{{{-1}} {{{{ \alpha} _{,i}} _{,j}}} {{{{ e} ^i} _I}} {{{{ e} ^j} _J}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^A}} {{{{ \bar{A}} _I} _J}} {{{{ e} ^a} _A}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _I} _J}} {{{{{ e} ^a} _A} _{,a}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{A}} _I} _J}} {{{{ \beta} ^A} _{,a}}} {{{{ e} ^a} _A}}} + {{{K}} {{\alpha}} \cdot {{{{ \bar{A}} _I} _J}}} + {{{-1}} {{W}} {{{ W} _{,i}}} {{{ \alpha} _{,j}}} {{{{ e} ^i} _I}} {{{{ e} ^j} _J}}} + {{{-1}} {{W}} {{{ W} _{,j}}} {{{ \alpha} _{,i}}} {{{{ e} ^i} _I}} {{{{ e} ^j} _J}}} + {{{\alpha}} \cdot {{{{ R} _I} _J}} {{{W}^{2}}}} + {{{{ \beta} ^A}} {{{{ e} ^a} _A}} {{{{{ \bar{A}} _I} _J} _{,a}}}} + {{{{{ \bar{A}} _A} _J}} {{{{ \beta} ^A} _{,i}}} {{{{ e} ^i} _I}}} + {{{{{ \bar{A}} _A} _I}} {{{{ \beta} ^A} _{,j}}} {{{{ e} ^j} _J}}} + {{{8}} \cdot {{\frac{1}{3}}} {{S}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ \bar{\gamma}} _I} _J}}} + {{{{ \beta} ^A}} {{{{ \bar{A}} _I} _N}} {{{{ e} ^a} _A}} {{{{ e} ^j} _J}} {{{{{ e} _j} ^N} _{,a}}}} + {{{{ \beta} ^A}} {{{{ \bar{A}} _M} _J}} {{{{ e} ^a} _A}} {{{{ e} ^i} _I}} {{{{{ e} _i} ^M} _{,a}}}} + {{{{ \beta} ^B}} {{{{ \bar{A}} _A} _J}} {{{{ e} _a} ^A}} {{{{ e} ^i} _I}} {{{{{ e} ^a} _B} _{,i}}}} + {{{{ \beta} ^B}} {{{{ \bar{A}} _A} _I}} {{{{ e} _a} ^A}} {{{{ e} ^j} _J}} {{{{{ e} ^a} _B} _{,j}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{\alpha}} \cdot {{{{ R} _A} _B}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} _I} _J}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \alpha} _{,a}}} {{{ \bar{\Lambda}} ^A}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \alpha} _{,a}}} {{{ \hat{\Gamma}} ^A}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{W}^{2}}}} + {{{\frac{1}{3}}} {{{ \alpha} _{,a}}} {{{ \mathcal{C}} ^A}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{W}^{2}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _A} _J}} {{{{ \bar{A}} _B} _I}} {{{{ \bar{\gamma}} ^A} ^B}}} + {{{-8}} {{\alpha}} \cdot {{\pi}} \cdot {{{{ S} _I} _J}} {{{W}^{2}}}} + {{{\frac{1}{3}}} {{{{ \alpha} _{,a}} _{,b}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{{ \bar{\gamma}} _I} _J} _{,b}}} {{{W}^{2}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ e} ^a} _A}} {{{{ e} ^i} _I}} {{{{{ \bar{\gamma}} _B} _J} _{,i}}} {{{W}^{2}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ e} ^a} _A}} {{{{ e} ^j} _J}} {{{{{ \bar{\gamma}} _B} _I} _{,j}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{{ \bar{A}} _B} _C} _{,a}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ e} ^a} _M}} {{{{ e} ^i} _I}} {{{{ e} ^j} _J}} {{{{{ e} _i} ^M} _{,j}}} {{{W}^{2}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} _C} _I}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^j} _J}} {{{{{ e} _b} ^C} _{,j}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} _M} _J}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^i} _I}} {{{{{ e} _i} ^M} _{,b}}} {{{W}^{2}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} _C} _J}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^i} _I}} {{{{{ e} _b} ^C} _{,i}}} {{{W}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} _I} _N}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^j} _J}} {{{{{ e} _j} ^N} _{,b}}} {{{W}^{2}}}} + {{{\frac{1}{2}}} {{{ \alpha} _{,a}}} {{{{ e} ^a} _N}} {{{{ e} ^i} _I}} {{{{ e} ^j} _J}} {{{{{ e} _j} ^N} _{,i}}} {{{W}^{2}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{W}} {{{ W} _{,a}}} {{{ \alpha} _{,b}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _D} _C}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{{ e} _b} ^D} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _E}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{{ e} _c} ^E} _{,a}}}} + {{{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^e} _E}} {{{{{ \bar{\gamma}} _A} _D} _{,e}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^d} _D}} {{{{{ \bar{\gamma}} _A} _E} _{,d}}}} + {{{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{{ \bar{\gamma}} _D} _E} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _A} _G}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{{ e} _e} ^G} _{,d}}}} + {{{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} ^e} _E}} {{{{{ e} _a} ^B} _{,e}}}} + {{{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} ^d} _D}} {{{{{ e} _d} ^C} _{,a}}}} + {{{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _A} _G}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{{ e} _d} ^G} _{,e}}}} + {{{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _G} _C}} {{{{ \bar{\gamma}} ^C} ^E}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} ^e} _E}} {{{{{ e} _e} ^G} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \beta} ^A}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^B} ^D}} {{{{ \bar{\gamma}} _I} _J}} {{{{ e} ^a} _A}} {{{{ e} ^d} _D}} {{{{{ e} _a} ^C} _{,d}}}}}$
${{{ \bar{\Lambda}} ^I} _{,t}} = {{{{\frac{1}{6}}} {{{ \beta} ^I}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{3}}} {{{ \beta} ^B}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ e} ^a} _A}} {{{{{{ e} ^b} _B} _{,a}} _{,b}}}} + {{{\frac{1}{3}}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{{ \beta} ^B} _{,a}} _{,b}}}} + {{{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ e} ^b} _B}} {{{{{ e} ^a} _A} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \beta} ^A} _{,b}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \bar{\gamma}} _{,b}}} {{{ \beta} ^B}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{\frac{1}{{\bar{\gamma}}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{6}}} {{{ \bar{\gamma}} _{,a}}} {{{ \bar{\gamma}} _{,b}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{\frac{1}{{\bar{\gamma}}^{2}}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{ \beta} ^C}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{{ \hat{\Gamma}} ^I} _A} _B}} {{{{{ e} ^c} _C} _{,c}}}} + {{{-2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \beta} ^C} _{,c}}} {{{{ e} ^c} _C}} {{{{{ \hat{\Gamma}} ^I} _A} _B}}} + {{{{{ \bar{\gamma}} ^A} ^B}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{{ \beta} ^I} _{,a}} _{,b}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^B}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ e} ^a} _A}} {{{{{ e} ^b} _B} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{{ \hat{\Gamma}} ^I} _B} _C}} {{\frac{1}{\bar{\gamma}}}}} + {{{-1}} \cdot {{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \beta} ^B} _{,b}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{2}}} {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{\gamma}} ^B} ^A}} {{{{ \beta} ^I} _{,b}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{\frac{1}{\bar{\gamma}}}}} + {{{{ \beta} ^J}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _i} ^I}} {{{{{{ e} ^i} _J} _{,a}} _{,b}}}} + {{{\frac{1}{2}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^J}} {{{{ \bar{\gamma}} ^B} ^A}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _i} ^I}} {{{{{ e} ^i} _J} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _A} _B}} {{{{ \bar{\gamma}} ^C} ^A}} {{{{ \bar{\gamma}} ^D} ^B}} {{{{{ \hat{\Gamma}} ^I} _C} _D}}} + {{{-2}} {{{ \alpha} _{,a}}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^A} ^C}} {{{{ \bar{\gamma}} ^I} ^B}} {{{{ e} ^a} _A}}} + {{{2}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \beta} ^C} _{,a}}} {{{{ e} ^a} _A}} {{{{{ \hat{\Gamma}} ^I} _B} _C}}} + {{{-1}} {{\alpha}} \cdot {{{ \bar{\gamma}} _{,a}}} {{{{ \bar{A}} _B} _C}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} ^I} ^C}} {{{{ e} ^a} _A}} {{\frac{1}{\bar{\gamma}}}}} + {{{2}} {{{ \beta} ^C}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ e} ^a} _A}} {{{{ e} _c} ^D}} {{{{{ \hat{\Gamma}} ^I} _B} _D}} {{{{{ e} ^c} _C} _{,a}}}} + {{{\alpha}} \cdot {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{{ \bar{A}} _B} _C} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{{{ \bar{\gamma}} _C} _D} _{,a}} _{,b}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \beta} ^D} _{,b}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{{ \bar{\gamma}} _C} _D} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \beta} ^D} _{,c}}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{{ \bar{\gamma}} _B} _D} _{,a}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^c} _C}} {{{{{ \bar{A}} _A} _B} _{,c}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} ^C} ^I}} {{{{ e} ^b} _B}} {{{{{ \bar{\gamma}} _A} _C} _{,b}}} {{{{{ e} ^d} _D} _{,d}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} ^C} ^I}} {{{{ \beta} ^D} _{,d}}} {{{{ e} ^b} _B}} {{{{ e} ^d} _D}} {{{{{ \bar{\gamma}} _A} _C} _{,b}}}} + {{{\alpha}} \cdot {{{{ \bar{A}} _B} _E}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{{ e} _c} ^E} _{,a}}}} + {{{\alpha}} \cdot {{{{ \bar{A}} _D} _C}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{{ e} _b} ^D} _{,a}}}} + {{{-1}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^b} _B}} {{{{{ \bar{\gamma}} _A} _C} _{,d}}} {{{{{ e} ^d} _D} _{,b}}}} + {{{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} ^D} ^E}} {{{{ e} ^a} _A}} {{{{{ \bar{\gamma}} _B} _D} _{,a}}} {{{{{ \hat{\Gamma}} ^I} _C} _E}}} + {{{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^c} _C}} {{{{{ \bar{\gamma}} _A} _B} _{,d}}} {{{{{ e} ^d} _D} _{,c}}}} + {{{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ e} ^a} _A}} {{{{ e} ^d} _D}} {{{{{{ \bar{\gamma}} _B} _C} _{,a}} _{,d}}}} + {{{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \beta} ^D} _{,a}}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{{ \bar{\gamma}} _B} _D} _{,c}}}} + {{{-1}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \beta} ^I} _{,c}}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{{ \bar{\gamma}} _A} _D} _{,b}}}} + {{{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \beta} ^D} _{,c}}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{{ \bar{\gamma}} _A} _B} _{,d}}}} + {{{-1}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} ^C} ^I}} {{{{ \beta} ^D} _{,c}}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{{ \bar{\gamma}} _A} _D} _{,b}}}} + {{{-1}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \beta} ^D} _{,b}}} {{{{ e} ^b} _B}} {{{{ e} ^d} _D}} {{{{{ \bar{\gamma}} _A} _C} _{,d}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} ^D} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{{ \bar{\gamma}} _B} _D} _{,c}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{2}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^B}} {{{{ \bar{\gamma}} ^C} ^A}} {{{{ \bar{\gamma}} ^D} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{{ \bar{\gamma}} _C} _D} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{2}}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^b} _F}} {{{{{ e} ^d} _D} _{,b}}} {{{{{ e} _d} ^F} _{,a}}}} + {{{\frac{1}{2}}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} _d} ^F}} {{{{{ \bar{\gamma}} _C} _F} _{,a}}} {{{{{ e} ^d} _D} _{,b}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{{{ e} _c} ^C} _{,a}} _{,b}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^d} _F}} {{{{{{ e} _d} ^F} _{,a}} _{,b}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^c} _F}} {{{{{ e} ^d} _D} _{,c}}} {{{{{ e} _d} ^F} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{ e} _d} ^F}} {{{{{ \bar{\gamma}} _B} _F} _{,a}}} {{{{{ e} ^d} _D} _{,c}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \beta} ^D} _{,c}}} {{{{ e} ^a} _A}} {{{{ e} ^c} _F}} {{{{ e} ^d} _D}} {{{{{ e} _d} ^F} _{,a}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} _E} _D}} {{{{ \beta} ^D} _{,c}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{{ e} _b} ^E} _{,a}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} _E} _D}} {{{{ \beta} ^D} _{,b}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{{ e} _c} ^E} _{,a}}}} + {{{\frac{1}{2}}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \beta} ^D} _{,b}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _F}} {{{{ e} ^d} _D}} {{{{{ e} _d} ^F} _{,a}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _A} _E}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{{ e} _b} ^E} _{,c}}}} + {{{-2}} {{\alpha}} \cdot {{{{ \bar{A}} _D} _B}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{{ e} _a} ^D} _{,c}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{{ e} _a} ^I} _{,b}}} {{{{{ e} ^d} _D} _{,d}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^C} ^I}} {{{{ e} ^b} _F}} {{{{ e} ^c} _C}} {{{{{ e} _c} ^F} _{,b}}} {{{{{ e} ^d} _D} _{,d}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^C} ^I}} {{{{ \beta} ^D} _{,d}}} {{{{ e} ^b} _F}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{{ e} _c} ^F} _{,b}}}} + {{{2}} \cdot {{\frac{1}{3}}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \beta} ^D} _{,d}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^d} _D}} {{{{{ e} _a} ^I} _{,b}}}} + {{{-1}} {{\alpha}} \cdot {{{{ \bar{A}} _A} _B}} {{{{ \bar{\gamma}} ^C} ^A}} {{{{ \bar{\gamma}} ^D} ^B}} {{{{ \bar{\gamma}} ^E} ^I}} {{{{ e} ^e} _E}} {{{{{ \bar{\gamma}} _C} _D} _{,e}}}} + {{{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{{ e} _a} ^C} _{,d}}} {{{{{ e} ^d} _D} _{,c}}}} + {{{-1}} {{{ \beta} ^J}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _i} ^I}} {{{{{ \bar{\gamma}} _A} _D} _{,b}}} {{{{{ e} ^i} _J} _{,c}}}} + {{{{ \beta} ^A}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{{{ e} _c} ^I} _{,a}} _{,d}}}} + {{{{ \beta} ^D}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{{ e} _b} ^I} _{,d}}} {{{{{ e} ^d} _D} _{,c}}}} + {{{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{{ \hat{\Gamma}} ^I} _C} _E}} {{{{{ e} _b} ^E} _{,a}}}} + {{{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{ e} _d} ^F}} {{{{{ \bar{\gamma}} _B} _F} _{,c}}} {{{{{ e} ^d} _D} _{,a}}}} + {{{-1}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^C} ^I}} {{{{ e} ^b} _F}} {{{{ e} ^c} _C}} {{{{{ e} ^d} _D} _{,c}}} {{{{{ e} _d} ^F} _{,b}}}} + {{{{ \beta} ^A}} {{{{ \bar{\gamma}} ^D} ^E}} {{{{ e} ^a} _A}} {{{{ e} ^d} _D}} {{{{{ \hat{\Gamma}} ^I} _G} _E}} {{{{{ e} _d} ^G} _{,a}}}} + {{{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^c} _F}} {{{{{ e} ^d} _D} _{,a}}} {{{{{ e} _d} ^F} _{,c}}}} + {{{-1}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{{ e} _c} ^I} _{,d}}} {{{{{ e} ^d} _D} _{,b}}}} + {{{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^d} _D}} {{{{{{ e} _b} ^D} _{,a}} _{,d}}}} + {{{-1}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} ^C} ^I}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _d} ^F}} {{{{{ \bar{\gamma}} _A} _F} _{,b}}} {{{{{ e} ^d} _D} _{,c}}}} + {{{-1}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{{ e} _a} ^B} _{,d}}} {{{{{ e} ^d} _D} _{,b}}}} + {{{-1}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \beta} ^I} _{,c}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{{ e} _a} ^C} _{,b}}}} + {{{{{ \bar{\gamma}} ^A} ^I}} {{{{ \beta} ^D} _{,a}}} {{{{ e} ^a} _A}} {{{{ e} ^c} _F}} {{{{ e} ^d} _D}} {{{{{ e} _d} ^F} _{,c}}}} + {{{-1}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} ^C} ^I}} {{{{ \bar{\gamma}} _E} _D}} {{{{ \beta} ^D} _{,c}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{{ e} _a} ^E} _{,b}}}} + {{{-1}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \beta} ^D} _{,b}}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{{ e} _c} ^I} _{,d}}}} + {{{-1}} {{{{ \bar{\gamma}} ^C} ^I}} {{{{ \beta} ^D} _{,c}}} {{{{ e} ^b} _F}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{{ e} _d} ^F} _{,b}}}} + {{{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} _E} _D}} {{{{ \beta} ^D} _{,a}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{{ e} _b} ^E} _{,c}}}} + {{{{{ \bar{\gamma}} ^B} ^C}} {{{{ \beta} ^D} _{,c}}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{{ e} _b} ^I} _{,d}}}} + {{{-1}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \beta} ^I} _{,c}}} {{{{ e} ^b} _F}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{{ e} _d} ^F} _{,b}}}} + {{{-1}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \beta} ^D} _{,b}}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^d} _D}} {{{{{ e} _a} ^B} _{,d}}}} + {{{{{ \bar{\gamma}} ^A} ^I}} {{{{ \beta} ^D} _{,c}}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{{ e} _a} ^C} _{,d}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{{ e} _b} ^I} _{,c}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{3}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^D} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^c} _F}} {{{{ e} ^d} _D}} {{{{{ e} _d} ^F} _{,c}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{2}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^B}} {{{{ \bar{\gamma}} ^D} ^I}} {{{{ e} ^a} _F}} {{{{ e} ^b} _B}} {{{{ e} ^d} _D}} {{{{{ e} _d} ^F} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{2}}} {{{ \bar{\gamma}} _{,a}}} {{{ \beta} ^B}} {{{{ \bar{\gamma}} ^C} ^A}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{{ e} _c} ^I} _{,b}}} {{\frac{1}{\bar{\gamma}}}}} + {{{\frac{1}{2}}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} _E} _F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _d} ^F}} {{{{{ e} _c} ^E} _{,a}}} {{{{{ e} ^d} _D} _{,b}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} _E} _F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _d} ^F}} {{{{{ e} _b} ^E} _{,a}}} {{{{{ e} ^d} _D} _{,c}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _A} _B}} {{{{ \bar{\gamma}} ^C} ^A}} {{{{ \bar{\gamma}} ^D} ^E}} {{{{ \bar{\gamma}} ^I} ^B}} {{{{ e} ^e} _E}} {{{{{ \bar{\gamma}} _C} _D} _{,e}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _A} _B}} {{{{ \bar{\gamma}} ^C} ^A}} {{{{ \bar{\gamma}} ^D} ^B}} {{{{ \bar{\gamma}} ^E} ^I}} {{{{ e} ^d} _D}} {{{{{ \bar{\gamma}} _C} _E} _{,d}}}} + {{{-1}} {{\alpha}} \cdot {{{{ \bar{A}} _G} _B}} {{{{ \bar{\gamma}} ^D} ^B}} {{{{ \bar{\gamma}} ^E} ^I}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{{ e} _d} ^G} _{,e}}}} + {{{-1}} {{\alpha}} \cdot {{{{ \bar{A}} _A} _B}} {{{{ \bar{\gamma}} ^C} ^A}} {{{{ \bar{\gamma}} ^E} ^I}} {{{{ e} ^c} _C}} {{{{ e} ^e} _E}} {{{{{ e} _c} ^B} _{,e}}}} + {{{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^I}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} _E} _F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _d} ^F}} {{{{{ e} _b} ^E} _{,c}}} {{{{{ e} ^d} _D} _{,a}}}} + {{{-1}} {{{ \beta} ^D}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ \bar{\gamma}} ^C} ^I}} {{{{ \bar{\gamma}} _E} _F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _d} ^F}} {{{{{ e} _a} ^E} _{,b}}} {{{{{ e} ^d} _D} _{,c}}}} + {{{-1}} {{{ \beta} ^J}} {{{{ \bar{\gamma}} ^A} ^B}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} _i} ^I}} {{{{{ e} _a} ^C} _{,b}}} {{{{{ e} ^i} _J} _{,c}}}} + {{{-1}} {{{ \beta} ^J}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ e} ^b} _F}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} _i} ^I}} {{{{{ e} _d} ^F} _{,b}}} {{{{{ e} ^i} _J} _{,c}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _G} _B}} {{{{ \bar{\gamma}} ^D} ^E}} {{{{ \bar{\gamma}} ^I} ^B}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{{ e} _d} ^G} _{,e}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _A} _B}} {{{{ \bar{\gamma}} ^C} ^A}} {{{{ \bar{\gamma}} ^D} ^B}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{{ e} _c} ^I} _{,d}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _A} _B}} {{{{ \bar{\gamma}} ^C} ^A}} {{{{ \bar{\gamma}} ^I} ^B}} {{{{ e} ^c} _C}} {{{{ e} ^e} _E}} {{{{{ e} _c} ^E} _{,e}}}} + {{{2}} {{\alpha}} \cdot {{{{ \bar{A}} _G} _B}} {{{{ \bar{\gamma}} ^D} ^B}} {{{{ \bar{\gamma}} ^E} ^I}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{{ e} _e} ^G} _{,d}}}} + {{{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{{ \bar{\gamma}} _C} _E} _{,a}}} {{{{{ \bar{\gamma}} _D} _F} _{,b}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ e} ^a} _A}} {{{{ e} ^f} _F}} {{{{{ \bar{\gamma}} _B} _D} _{,f}}} {{{{{ \bar{\gamma}} _C} _E} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} ^D} ^E}} {{{{ \bar{\gamma}} ^F} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{{ \bar{\gamma}} _B} _E} _{,c}}} {{{{{ \bar{\gamma}} _D} _F} _{,a}}}} + {{{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^f} _F}} {{{{{ \bar{\gamma}} _K} _E} _{,a}}} {{{{{ e} _f} ^K} _{,b}}}} + {{{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _F}} {{{{ e} ^d} _D}} {{{{{ \bar{\gamma}} _C} _E} _{,a}}} {{{{{ e} _d} ^I} _{,b}}}} + {{{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{{ \bar{\gamma}} _D} _F} _{,b}}} {{{{{ e} _c} ^F} _{,a}}}} + {{{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^e} _E}} {{{{{ \bar{\gamma}} _H} _F} _{,b}}} {{{{{ e} _e} ^H} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} ^D} ^E}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ δ} ^K} _K}} {{{{{ \bar{\gamma}} _B} _E} _{,c}}} {{{{{ e} _d} ^I} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ e} ^a} _A}} {{{{ e} ^e} _E}} {{{{ e} ^f} _F}} {{{{{ \bar{\gamma}} _B} _K} _{,f}}} {{{{{ e} _e} ^K} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} ^F} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{{ \bar{\gamma}} _D} _F} _{,a}}} {{{{{ e} _b} ^D} _{,c}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{ e} ^f} _F}} {{{{{ \bar{\gamma}} _E} _D} _{,f}}} {{{{{ e} _c} ^I} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^f} _F}} {{{{{ \bar{\gamma}} _C} _E} _{,a}}} {{{{{ e} _b} ^C} _{,f}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^D} ^E}} {{{{ \bar{\gamma}} ^F} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^c} _H}} {{{{ e} ^e} _E}} {{{{{ \bar{\gamma}} _D} _F} _{,a}}} {{{{{ e} _e} ^H} _{,c}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ e} ^a} _A}} {{{{ e} ^d} _D}} {{{{ e} ^f} _F}} {{{{{ \bar{\gamma}} _C} _E} _{,a}}} {{{{{ e} _d} ^I} _{,f}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} ^F} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{ e} ^f} _F}} {{{{{ \bar{\gamma}} _B} _K} _{,c}}} {{{{{ e} _f} ^K} _{,a}}}} + {{{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ e} ^a} _A}} {{{{ e} ^b} _F}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{{ e} _c} ^F} _{,a}}} {{{{{ e} _d} ^I} _{,b}}}} + {{{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _F}} {{{{ e} ^d} _H}} {{{{ e} ^e} _E}} {{{{{ e} _d} ^I} _{,b}}} {{{{{ e} _e} ^H} _{,a}}}} + {{{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ \bar{\gamma}} _H} _K}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^e} _E}} {{{{ e} ^f} _F}} {{{{{ e} _e} ^H} _{,a}}} {{{{{ e} _f} ^K} _{,b}}}} + {{{\frac{1}{2}}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _K}} {{{{ e} ^f} _F}} {{{{{ e} _c} ^F} _{,a}}} {{{{{ e} _f} ^K} _{,b}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ δ} ^K} _K}} {{{{{ e} _b} ^D} _{,c}}} {{{{{ e} _d} ^I} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ e} ^a} _A}} {{{{ e} ^d} _K}} {{{{ e} ^e} _E}} {{{{ e} ^f} _F}} {{{{{ e} _d} ^I} _{,f}}} {{{{{ e} _e} ^K} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^D} ^E}} {{{{ e} ^a} _A}} {{{{ e} ^c} _H}} {{{{ e} ^d} _D}} {{{{ e} ^e} _E}} {{{{ δ} ^K} _K}} {{{{{ e} _d} ^I} _{,a}}} {{{{{ e} _e} ^H} _{,c}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ e} ^a} _A}} {{{{ e} ^b} _E}} {{{{ e} ^c} _C}} {{{{ e} ^f} _F}} {{{{{ e} _b} ^C} _{,f}}} {{{{{ e} _c} ^I} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^C} ^D}} {{{{ e} ^a} _A}} {{{{ e} ^c} _C}} {{{{ e} ^d} _D}} {{{{ e} ^f} _F}} {{{{{ e} _c} ^I} _{,a}}} {{{{{ e} _d} ^F} _{,f}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^C}} {{{{ \bar{\gamma}} ^F} ^I}} {{{{ \bar{\gamma}} _D} _K}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^c} _C}} {{{{ e} ^f} _F}} {{{{{ e} _b} ^D} _{,c}}} {{{{{ e} _f} ^K} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^F} ^I}} {{{{ e} ^a} _A}} {{{{ e} ^c} _H}} {{{{ e} ^e} _K}} {{{{ e} ^f} _F}} {{{{{ e} _e} ^H} _{,c}}} {{{{{ e} _f} ^K} _{,a}}}} + {{{-1}} {{{ \beta} ^A}} {{{{ \bar{\gamma}} ^B} ^I}} {{{{ \bar{\gamma}} ^E} ^F}} {{{{ \bar{\gamma}} _C} _K}} {{{{ e} ^a} _A}} {{{{ e} ^b} _B}} {{{{ e} ^e} _E}} {{{{ e} ^f} _F}} {{{{{ e} _b} ^C} _{,f}}} {{{{{ e} _e} ^K} _{,a}}}}}$

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