${{{ g} _u} _v} = {\left[\begin{array}{cc} {-{{\alpha}^{2}}} + {{{{ \beta} ^k}} {{{ \beta} _k}}}& { \beta} _j\\ { \beta} _i& {{ \gamma} _i} _j\end{array}\right]}$
${{{ \Lambda} ^u} _v} = {\left[\begin{array}{cc} \Gamma& {-{\Gamma}} {{B}} {{{ n} _j}}\\ {-{\Gamma}} {{B}} {{{ n} ^i}}& {{{ \delta} ^i} _j} + {{{\left({{\Gamma}{-{1}}}\right)}} {{{ n} ^i}} {{{ n} _j}}}\end{array}\right]}$
${{{{ g'} _a} _b} = {{{{{ \Lambda} ^u} _a}} {{{{ g} _u} _v}} {{{{ \Lambda} ^v} _b}}}} = {{{\left[\begin{array}{cc} \Gamma& {-{\Gamma}} {{B}} {{{ n} ^i}}\\ {-{\Gamma}} {{B}} {{{ n} _a}}& {{{ \delta} ^i} _a} + {{{\left({{\Gamma}{-{1}}}\right)}} {{{ n} ^i}} {{{ n} _a}}}\end{array}\right]}} {{\left[\begin{array}{cc} {-{{\alpha}^{2}}} + {{{{ \beta} ^k}} {{{ \beta} _k}}}& { \beta} _j\\ { \beta} _i& {{ \gamma} _i} _j\end{array}\right]}} {{\left[\begin{array}{cc} \Gamma& {-{\Gamma}} {{B}} {{{ n} _b}}\\ {-{\Gamma}} {{B}} {{{ n} ^j}}& {{{ \delta} ^j} _b} + {{{\left({{\Gamma}{-{1}}}\right)}} {{{ n} ^j}} {{{ n} _b}}}\end{array}\right]}}}$
${{{ g'} _a} _b} = {\left[\begin{array}{cc} {-{{{{\Gamma}^{2}}} {{{\alpha}^{2}}}}} + {{{{ \beta} ^k}} {{{ \beta} _k}} {{{\Gamma}^{2}}}}{-{{{B}} {{{ \beta} _j}} {{{ n} ^j}} {{{\Gamma}^{2}}}}}{-{{{B}} {{{ \beta} _i}} {{{ n} ^i}} {{{\Gamma}^{2}}}}} + {{{{ n} ^i}} {{{ n} ^j}} {{{{ \gamma} _i} _j}} {{{B}^{2}}} {{{\Gamma}^{2}}}}& {{{B}} {{{ n} _b}} {{{\Gamma}^{2}}} {{{\alpha}^{2}}}} + {{{\Gamma}} \cdot {{{ \beta} _j}} {{{{ \delta} ^j} _b}}}{-{{{\Gamma}} \cdot {{{ \beta} _j}} {{{ n} _b}} {{{ n} ^j}}}} + {{{{ \beta} _j}} {{{ n} _b}} {{{ n} ^j}} {{{\Gamma}^{2}}}} + {{{{ \beta} _i}} {{{ n} _b}} {{{ n} ^i}} {{{B}^{2}}} {{{\Gamma}^{2}}}}{-{{{B}} {{\Gamma}} \cdot {{{ n} ^i}} {{{{ \delta} ^j} _b}} {{{{ \gamma} _i} _j}}}}{-{{{B}} {{{ \beta} _k}} {{{ \beta} ^k}} {{{ n} _b}} {{{\Gamma}^{2}}}}} + {{{B}} {{\Gamma}} \cdot {{{ n} _b}} {{{ n} ^i}} {{{ n} ^j}} {{{{ \gamma} _i} _j}}}{-{{{B}} {{{ n} _b}} {{{ n} ^i}} {{{ n} ^j}} {{{{ \gamma} _i} _j}} {{{\Gamma}^{2}}}}}\\ {{{B}} {{{ n} _a}} {{{\Gamma}^{2}}} {{{\alpha}^{2}}}} + {{{\Gamma}} \cdot {{{ \beta} _i}} {{{{ \delta} ^i} _a}}}{-{{{\Gamma}} \cdot {{{ \beta} _i}} {{{ n} _a}} {{{ n} ^i}}}} + {{{{ \beta} _i}} {{{ n} _a}} {{{ n} ^i}} {{{\Gamma}^{2}}}} + {{{{ \beta} _j}} {{{ n} _a}} {{{ n} ^j}} {{{B}^{2}}} {{{\Gamma}^{2}}}}{-{{{B}} {{\Gamma}} \cdot {{{ n} ^j}} {{{{ \delta} ^i} _a}} {{{{ \gamma} _i} _j}}}}{-{{{B}} {{{ \beta} _k}} {{{ \beta} ^k}} {{{ n} _a}} {{{\Gamma}^{2}}}}} + {{{B}} {{\Gamma}} \cdot {{{ n} _a}} {{{ n} ^i}} {{{ n} ^j}} {{{{ \gamma} _i} _j}}}{-{{{B}} {{{ n} _a}} {{{ n} ^i}} {{{ n} ^j}} {{{{ \gamma} _i} _j}} {{{\Gamma}^{2}}}}}& {{{{{ \delta} ^i} _a}} {{{{ \delta} ^j} _b}} {{{{ \gamma} _i} _j}}}{-{{{{ n} _b}} {{{ n} ^j}} {{{{ \delta} ^i} _a}} {{{{ \gamma} _i} _j}}}}{-{{{{ n} _a}} {{{ n} _b}} {{{B}^{2}}} {{{\Gamma}^{2}}} {{{\alpha}^{2}}}}}{-{{{{ n} _a}} {{{ n} ^i}} {{{{ \delta} ^j} _b}} {{{{ \gamma} _i} _j}}}}{-{{{B}} {{\Gamma}} \cdot {{{ \beta} _j}} {{{ n} _a}} {{{{ \delta} ^j} _b}}}}{-{{{B}} {{\Gamma}} \cdot {{{ \beta} _i}} {{{ n} _b}} {{{{ \delta} ^i} _a}}}} + {{{\Gamma}} \cdot {{{ n} _b}} {{{ n} ^j}} {{{{ \delta} ^i} _a}} {{{{ \gamma} _i} _j}}} + {{{\Gamma}} \cdot {{{ n} _a}} {{{ n} ^i}} {{{{ \delta} ^j} _b}} {{{{ \gamma} _i} _j}}} + {{{{ \beta} ^k}} {{{ \beta} _k}} {{{ n} _a}} {{{ n} _b}} {{{B}^{2}}} {{{\Gamma}^{2}}}} + {{{{ n} _a}} {{{ n} _b}} {{{ n} ^i}} {{{ n} ^j}} {{{{ \gamma} _i} _j}}} + {{{B}} {{\Gamma}} \cdot {{{ \beta} _i}} {{{ n} _a}} {{{ n} _b}} {{{ n} ^i}}} + {{{B}} {{\Gamma}} \cdot {{{ \beta} _j}} {{{ n} _a}} {{{ n} _b}} {{{ n} ^j}}}{-{{{B}} {{{ \beta} _j}} {{{ n} _a}} {{{ n} _b}} {{{ n} ^j}} {{{\Gamma}^{2}}}}}{-{{{B}} {{{ \beta} _i}} {{{ n} _a}} {{{ n} _b}} {{{ n} ^i}} {{{\Gamma}^{2}}}}} + {{{{ n} _a}} {{{ n} _b}} {{{ n} ^i}} {{{ n} ^j}} {{{{ \gamma} _i} _j}} {{{\Gamma}^{2}}}}{-{{{2}} {{\Gamma}} \cdot {{{ n} _a}} {{{ n} _b}} {{{ n} ^i}} {{{ n} ^j}} {{{{ \gamma} _i} _j}}}}\end{array}\right]}$
${{{ g'} _a} _b} = {\left[\begin{array}{cc} {{{{B}^{2}}} {{{\Gamma}^{2}}}}{-{{{{\Gamma}^{2}}} {{{\alpha}^{2}}}}} + {{{{\Gamma}^{2}}} {{{\beta}^{2}}}}{-{{{2}} {{B}} {{{ \beta} _k}} {{{ n} ^k}} {{{\Gamma}^{2}}}}}& {{{\Gamma}} \cdot {{{ \beta} _b}}}{-{{{B}} {{{ n} _b}} {{{\Gamma}^{2}}}}} + {{{B}} {{{ n} _b}} {{{\Gamma}^{2}}} {{{\alpha}^{2}}}}{-{{{B}} {{{ n} _b}} {{{\Gamma}^{2}}} {{{\beta}^{2}}}}}{-{{{\Gamma}} \cdot {{{ \beta} _k}} {{{ n} _b}} {{{ n} ^k}}}} + {{{{ \beta} _k}} {{{ n} _b}} {{{ n} ^k}} {{{\Gamma}^{2}}}} + {{{{ \beta} _k}} {{{ n} _b}} {{{ n} ^k}} {{{B}^{2}}} {{{\Gamma}^{2}}}}\\ {{{\Gamma}} \cdot {{{ \beta} _a}}}{-{{{B}} {{{ n} _a}} {{{\Gamma}^{2}}}}} + {{{B}} {{{ n} _a}} {{{\Gamma}^{2}}} {{{\alpha}^{2}}}}{-{{{B}} {{{ n} _a}} {{{\Gamma}^{2}}} {{{\beta}^{2}}}}}{-{{{\Gamma}} \cdot {{{ \beta} _k}} {{{ n} _a}} {{{ n} ^k}}}} + {{{{ \beta} _k}} {{{ n} _a}} {{{ n} ^k}} {{{\Gamma}^{2}}}} + {{{{ \beta} _k}} {{{ n} _a}} {{{ n} ^k}} {{{B}^{2}}} {{{\Gamma}^{2}}}}& {{{ \gamma} _a} _b}{-{{{{ n} _a}} {{{ n} _b}}}} + {{{{ n} _a}} {{{ n} _b}} {{{\Gamma}^{2}}}}{-{{{B}} {{\Gamma}} \cdot {{{ \beta} _b}} {{{ n} _a}}}}{-{{{B}} {{\Gamma}} \cdot {{{ \beta} _a}} {{{ n} _b}}}}{-{{{{ n} _a}} {{{ n} _b}} {{{B}^{2}}} {{{\Gamma}^{2}}} {{{\alpha}^{2}}}}} + {{{{ n} _a}} {{{ n} _b}} {{{B}^{2}}} {{{\Gamma}^{2}}} {{{\beta}^{2}}}} + {{{2}} {{B}} {{\Gamma}} \cdot {{{ \beta} _k}} {{{ n} _a}} {{{ n} _b}} {{{ n} ^k}}}{-{{{2}} {{B}} {{{ \beta} _k}} {{{ n} _a}} {{{ n} _b}} {{{ n} ^k}} {{{\Gamma}^{2}}}}}\end{array}\right]}$
Let ${\Gamma} = {\frac{1}{\sqrt{{1}{-{{B}^{2}}}}}}$
${{{ g'} _a} _b} = {\left[\begin{array}{cc} \frac{{{B}^{2}}{-{{\alpha}^{2}}} + {{\beta}^{2}}{-{{{2}} {{B}} {{{ \beta} _k}} {{{ n} ^k}}}}}{{1}{-{{B}^{2}}}}& \frac{{-{{{B}} {{{ n} _b}}}} + {{{{ \beta} _b}} {{\sqrt{{1}{-{{B}^{2}}}}}}} + {{{B}} {{{ n} _b}} {{{\alpha}^{2}}}}{-{{{B}} {{{ n} _b}} {{{\beta}^{2}}}}} + {{{{ \beta} _k}} {{{ n} _b}} {{{ n} ^k}}} + {{{{ \beta} _k}} {{{ n} _b}} {{{ n} ^k}} {{{B}^{2}}}}{-{{{{ \beta} _k}} {{{ n} _b}} {{{ n} ^k}} {{\sqrt{{1}{-{{B}^{2}}}}}}}}}{{1}{-{{B}^{2}}}}\\ \frac{{-{{{B}} {{{ n} _a}}}} + {{{{ \beta} _a}} {{\sqrt{{1}{-{{B}^{2}}}}}}} + {{{B}} {{{ n} _a}} {{{\alpha}^{2}}}}{-{{{B}} {{{ n} _a}} {{{\beta}^{2}}}}} + {{{{ \beta} _k}} {{{ n} _a}} {{{ n} ^k}}} + {{{{ \beta} _k}} {{{ n} _a}} {{{ n} ^k}} {{{B}^{2}}}}{-{{{{ \beta} _k}} {{{ n} _a}} {{{ n} ^k}} {{\sqrt{{1}{-{{B}^{2}}}}}}}}}{{1}{-{{B}^{2}}}}& \frac{{{{ \gamma} _a} _b}{-{{{{{ \gamma} _a} _b}} {{{B}^{2}}}}} + {{{{ n} _a}} {{{ n} _b}} {{{B}^{2}}}}{-{{{{ n} _a}} {{{ n} _b}} {{{B}^{2}}} {{{\alpha}^{2}}}}} + {{{{ n} _a}} {{{ n} _b}} {{{B}^{2}}} {{{\beta}^{2}}}}{-{{{B}} {{{ \beta} _a}} {{{ n} _b}} {{\sqrt{{1}{-{{B}^{2}}}}}}}}{-{{{B}} {{{ \beta} _b}} {{{ n} _a}} {{\sqrt{{1}{-{{B}^{2}}}}}}}}{-{{{2}} {{B}} {{{ \beta} _k}} {{{ n} _a}} {{{ n} _b}} {{{ n} ^k}}}} + {{{2}} {{B}} {{{ \beta} _k}} {{{ n} _a}} {{{ n} _b}} {{{ n} ^k}} {{\sqrt{{1}{-{{B}^{2}}}}}}}}{{1}{-{{B}^{2}}}}\end{array}\right]}$
Let ${{ n} ^i} = {{{\zeta}} \cdot {{{ \beta} ^i}}}$
${{{ g'} _a} _b} = {\left[\begin{array}{cc} \frac{{{B}^{2}}{-{{\alpha}^{2}}} + {{\beta}^{2}}{-{{{2}} {{B}} {{\zeta}} \cdot {{{\beta}^{2}}}}}}{{1}{-{{B}^{2}}}}& \frac{{{{ \beta} _b}} {{\left({{{{{\beta}^{2}}} {{{\zeta}^{2}}}} + {\sqrt{{1}{-{{B}^{2}}}}}{-{{{B}} {{\zeta}}}} + {{{{B}^{2}}} {{{\beta}^{2}}} {{{\zeta}^{2}}}}{-{{{{\beta}^{2}}} {{{\zeta}^{2}}} {{\sqrt{{1}{-{{B}^{2}}}}}}}} + {{{B}} {{\zeta}} \cdot {{{\alpha}^{2}}}}{-{{{B}} {{\zeta}} \cdot {{{\beta}^{2}}}}}}\right)}}}{{1}{-{{B}^{2}}}}\\ \frac{{{{ \beta} _a}} {{\left({{{{{\beta}^{2}}} {{{\zeta}^{2}}}} + {\sqrt{{1}{-{{B}^{2}}}}}{-{{{B}} {{\zeta}}}} + {{{{B}^{2}}} {{{\beta}^{2}}} {{{\zeta}^{2}}}}{-{{{{\beta}^{2}}} {{{\zeta}^{2}}} {{\sqrt{{1}{-{{B}^{2}}}}}}}} + {{{B}} {{\zeta}} \cdot {{{\alpha}^{2}}}}{-{{{B}} {{\zeta}} \cdot {{{\beta}^{2}}}}}}\right)}}}{{1}{-{{B}^{2}}}}& \frac{{{{ \gamma} _a} _b}{-{{{{{ \gamma} _a} _b}} {{{B}^{2}}}}} + {{{{ \beta} _a}} {{{ \beta} _b}} {{{B}^{2}}} {{{\zeta}^{2}}}}{-{{{{ \beta} _a}} {{{ \beta} _b}} {{{B}^{2}}} {{{\alpha}^{2}}} {{{\zeta}^{2}}}}} + {{{{ \beta} _a}} {{{ \beta} _b}} {{{B}^{2}}} {{{\beta}^{2}}} {{{\zeta}^{2}}}}{-{{{2}} {{B}} {{\zeta}} \cdot {{{ \beta} _a}} {{{ \beta} _b}} {{\sqrt{{1}{-{{B}^{2}}}}}}}}{-{{{2}} {{B}} {{{ \beta} _a}} {{{ \beta} _b}} {{{\beta}^{2}}} {{{\zeta}^{3}}}}} + {{{2}} {{B}} {{{ \beta} _a}} {{{ \beta} _b}} {{{\beta}^{2}}} {{{\zeta}^{3}}} {{\sqrt{{1}{-{{B}^{2}}}}}}}}{{1}{-{{B}^{2}}}}\end{array}\right]}$