${{{ \eta} _T} _T} = {-1}$; ${{{ \eta} _X} _X} = {1}$; ${{{ \eta} _Y} _Y} = {1}$; ${{{ \eta} _Z} _Z} = {1}$
${b} = {\frac{{{9.8}} {{m}}}{{s}^{2}}}$
${b} = {\frac{1.0903970549325\cdot{10^{-16}}}{m}}$
${{{ e} _u} ^I} = {\overset{u\downarrow I\rightarrow}{\left[\begin{array}{cccc} \sqrt{{1}{-{{{2}} {{b}} {{z}}}}}& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}$
metric:
${{{ g} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} {-{1}} + {{{2}} {{b}} {{z}}}& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}$
metric inverse:
${{{ g} ^a} ^b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} \frac{1}{{-{1}} + {{{2}} {{b}} {{z}}}}& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}$
metric derivative:
${{{{ g} _a} _b} _{,c}} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& {{2}} {{b}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{matrix}\right]}}$
1st kind Christoffel:
${{{{ \Gamma} _a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& b\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ b& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} -{b}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{matrix}\right]}}$
connection coefficients / 2nd kind Christoffel:
${{{{ \Gamma} ^a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \left[\begin{array}{cccc} 0& 0& 0& \frac{b}{{-{1}} + {{{2}} {{b}} {{z}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{b}{{-{1}} + {{{2}} {{b}} {{z}}}}& 0& 0& 0\end{array}\right] \\ \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right] \\ \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right] \\ \left[\begin{array}{cccc} -{b}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\end{matrix}\right]}}$
connection coefficients derivative:
${{{{{ \Gamma} ^a} _b} _c} _{,d}} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& -{\frac{{{2}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}}\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& -{\frac{{{2}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$
connection coefficients squared:
${{{{{{ \Gamma} ^a} _e} _c}} {{{{{ \Gamma} ^e} _b} _d}}} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} \frac{{b}^{2}}{{1}{-{{{2}} {{b}} {{z}}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{b}^{2}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{{b}^{2}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& \frac{{b}^{2}}{{1}{-{{{2}} {{b}} {{z}}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} \frac{{b}^{2}}{{1}{-{{{2}} {{b}} {{z}}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$
Riemann curvature, $\sharp\flat\flat\flat$:
${{{{{ R} ^a} _b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& \frac{{b}^{2}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ -{\frac{{b}^{2}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}}& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& \frac{{b}^{2}}{{1}{-{{{2}} {{b}} {{z}}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ -{\frac{{b}^{2}}{{1}{-{{{2}} {{b}} {{z}}}}}}& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$
Riemann curvature, $\sharp\sharp\flat\flat$:
${{{{{ R} ^a} ^b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& \frac{{b}^{2}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ -{\frac{{b}^{2}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}}& 0& 0& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& 0& 0& \frac{{b}^{2}}{{-{1}}{-{{{4}} {{{b}^{2}}} {{{z}^{2}}}}} + {{{4}} {{b}} {{z}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{{b}^{2}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\end{array}\right]}}$
Ricci curvature, $\sharp\flat$:
${{{ R} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} \frac{{b}^{2}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{b}^{2}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}\end{array}\right]}}$
Gaussian curvature:
${R} = {\frac{{{2}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}}$
trace-free Ricci, $\sharp\flat$:
${{{ {(R^{TF})}} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} \frac{{{0.5}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}& 0& 0& 0\\ 0& -{\frac{{{0.5}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}}& 0& 0\\ 0& 0& -{\frac{{{0.5}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}}& 0\\ 0& 0& 0& \frac{{{0.5}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}\end{array}\right]}}$
Einstein / trace-reversed Ricci curvature, $\sharp\flat$:
${{{ G} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& -{\frac{{b}^{2}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}}& 0& 0\\ 0& 0& -{\frac{{b}^{2}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}}& 0\\ 0& 0& 0& 0\end{array}\right]}}$
Schouten, $\sharp\flat$:
${{{ P} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} \frac{{{0.33333333333333}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}& 0& 0& 0\\ 0& -{\frac{{{0.16666666666667}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}}& 0& 0\\ 0& 0& -{\frac{{{0.16666666666667}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}}& 0\\ 0& 0& 0& \frac{{{0.33333333333333}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}\end{array}\right]}}$
Weyl, $\sharp\sharp\flat\flat$:
${{{{{ C} ^a} ^b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& -{\frac{{{0.16666666666667}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}}& 0& 0\\ \frac{{{0.16666666666667}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& -{\frac{{{0.16666666666667}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}}& 0\\ 0& 0& 0& 0\\ \frac{{{0.16666666666667}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& \frac{{{0.33333333333333}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ -{\frac{{{0.33333333333333}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}}& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& \frac{{{0.16666666666667}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}& 0& 0\\ -{\frac{{{0.16666666666667}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{{{0.33333333333333}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}& 0\\ 0& -{\frac{{{0.33333333333333}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{\frac{{{0.16666666666667}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}}\\ 0& 0& 0& 0\\ 0& \frac{{{0.16666666666667}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& \frac{{{0.16666666666667}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}& 0\\ 0& 0& 0& 0\\ -{\frac{{{0.16666666666667}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{\frac{{{0.33333333333333}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}}& 0\\ 0& \frac{{{0.33333333333333}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& -{\frac{{{0.16666666666667}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}}\\ 0& 0& \frac{{{0.16666666666667}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& \frac{{b}^{2}}{{-{3}}{-{{{12}} {{{b}^{2}}} {{{z}^{2}}}}} + {{{12}} {{b}} {{z}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{{{0.33333333333333}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{{{0.16666666666667}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}\\ 0& 0& 0& 0\\ 0& -{\frac{{{0.16666666666667}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{{0.16666666666667}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}\\ 0& 0& -{\frac{{{0.16666666666667}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}}& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$
Weyl, $\flat\flat\flat\flat$:
${{{{{ C} _a} _b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& \frac{{b}^{2}}{{6}{-{{{12}} {{b}} {{z}}}}}& 0& 0\\ \frac{{b}^{2}}{{-{6}} + {{{12}} {{b}} {{z}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& \frac{{b}^{2}}{{6}{-{{{12}} {{b}} {{z}}}}}& 0\\ 0& 0& 0& 0\\ \frac{{b}^{2}}{{-{6}} + {{{12}} {{b}} {{z}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& \frac{{b}^{2}}{{-{3}} + {{{6}} {{b}} {{z}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{{b}^{2}}{{3}{-{{{6}} {{b}} {{z}}}}}& 0& 0& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& \frac{{b}^{2}}{{-{6}} + {{{12}} {{b}} {{z}}}}& 0& 0\\ \frac{{b}^{2}}{{6}{-{{{12}} {{b}} {{z}}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{{{0.33333333333333}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}& 0\\ 0& -{\frac{{{0.33333333333333}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{\frac{{{0.16666666666667}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}}\\ 0& 0& 0& 0\\ 0& \frac{{{0.16666666666667}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}& 0& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& 0& \frac{{b}^{2}}{{-{6}} + {{{12}} {{b}} {{z}}}}& 0\\ 0& 0& 0& 0\\ \frac{{b}^{2}}{{6}{-{{{12}} {{b}} {{z}}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{\frac{{{0.33333333333333}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}}& 0\\ 0& \frac{{{0.33333333333333}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& -{\frac{{{0.16666666666667}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}}\\ 0& 0& \frac{{{0.16666666666667}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& 0& 0& \frac{{b}^{2}}{{3}{-{{{6}} {{b}} {{z}}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{{b}^{2}}{{-{3}} + {{{6}} {{b}} {{z}}}}& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{{{0.16666666666667}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}\\ 0& 0& 0& 0\\ 0& -{\frac{{{0.16666666666667}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}}& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{{0.16666666666667}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}\\ 0& 0& -{\frac{{{0.16666666666667}} {{{b}^{2}}}}{{1} + {{{4}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{4}} {{b}} {{z}}}}}}& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\end{array}\right]}}$
Plebanski, $\sharp\sharp\flat\flat$:
${{{{{ P} ^a} ^b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& -{\frac{{{0.083333333333333}} {{{b}^{4}}}}{{1} + {{{16}} {{{b}^{4}}} {{{z}^{4}}}}{-{{{32}} {{{b}^{3}}} {{{z}^{3}}}}} + {{{24}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{8}} {{b}} {{z}}}}}}& 0& 0\\ \frac{{{0.083333333333333}} {{{b}^{4}}}}{{1} + {{{16}} {{{b}^{4}}} {{{z}^{4}}}}{-{{{32}} {{{b}^{3}}} {{{z}^{3}}}}} + {{{24}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{8}} {{b}} {{z}}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& -{\frac{{{0.083333333333333}} {{{b}^{4}}}}{{1} + {{{16}} {{{b}^{4}}} {{{z}^{4}}}}{-{{{32}} {{{b}^{3}}} {{{z}^{3}}}}} + {{{24}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{8}} {{b}} {{z}}}}}}& 0\\ 0& 0& 0& 0\\ \frac{{{0.083333333333333}} {{{b}^{4}}}}{{1} + {{{16}} {{{b}^{4}}} {{{z}^{4}}}}{-{{{32}} {{{b}^{3}}} {{{z}^{3}}}}} + {{{24}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{8}} {{b}} {{z}}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& \frac{{{0.16666666666667}} {{{b}^{4}}}}{{1} + {{{16}} {{{b}^{4}}} {{{z}^{4}}}}{-{{{32}} {{{b}^{3}}} {{{z}^{3}}}}} + {{{24}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{8}} {{b}} {{z}}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ -{\frac{{{0.16666666666667}} {{{b}^{4}}}}{{1} + {{{16}} {{{b}^{4}}} {{{z}^{4}}}}{-{{{32}} {{{b}^{3}}} {{{z}^{3}}}}} + {{{24}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{8}} {{b}} {{z}}}}}}& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& \frac{{{0.083333333333333}} {{{b}^{4}}}}{{1} + {{{16}} {{{b}^{4}}} {{{z}^{4}}}}{-{{{32}} {{{b}^{3}}} {{{z}^{3}}}}} + {{{24}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{8}} {{b}} {{z}}}}}& 0& 0\\ -{\frac{{{0.083333333333333}} {{{b}^{4}}}}{{1} + {{{16}} {{{b}^{4}}} {{{z}^{4}}}}{-{{{32}} {{{b}^{3}}} {{{z}^{3}}}}} + {{{24}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{8}} {{b}} {{z}}}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{{{0.16666666666667}} {{{b}^{4}}}}{{1} + {{{16}} {{{b}^{4}}} {{{z}^{4}}}}{-{{{32}} {{{b}^{3}}} {{{z}^{3}}}}} + {{{24}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{8}} {{b}} {{z}}}}}& 0\\ 0& -{\frac{{{0.16666666666667}} {{{b}^{4}}}}{{1} + {{{16}} {{{b}^{4}}} {{{z}^{4}}}}{-{{{32}} {{{b}^{3}}} {{{z}^{3}}}}} + {{{24}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{8}} {{b}} {{z}}}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{\frac{{{0.083333333333333}} {{{b}^{4}}}}{{1} + {{{16}} {{{b}^{4}}} {{{z}^{4}}}}{-{{{32}} {{{b}^{3}}} {{{z}^{3}}}}} + {{{24}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{8}} {{b}} {{z}}}}}}\\ 0& 0& 0& 0\\ 0& \frac{{{0.083333333333333}} {{{b}^{4}}}}{{1} + {{{16}} {{{b}^{4}}} {{{z}^{4}}}}{-{{{32}} {{{b}^{3}}} {{{z}^{3}}}}} + {{{24}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{8}} {{b}} {{z}}}}}& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& \frac{{{0.083333333333333}} {{{b}^{4}}}}{{1} + {{{16}} {{{b}^{4}}} {{{z}^{4}}}}{-{{{32}} {{{b}^{3}}} {{{z}^{3}}}}} + {{{24}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{8}} {{b}} {{z}}}}}& 0\\ 0& 0& 0& 0\\ -{\frac{{{0.083333333333333}} {{{b}^{4}}}}{{1} + {{{16}} {{{b}^{4}}} {{{z}^{4}}}}{-{{{32}} {{{b}^{3}}} {{{z}^{3}}}}} + {{{24}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{8}} {{b}} {{z}}}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{\frac{{{0.16666666666667}} {{{b}^{4}}}}{{1} + {{{16}} {{{b}^{4}}} {{{z}^{4}}}}{-{{{32}} {{{b}^{3}}} {{{z}^{3}}}}} + {{{24}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{8}} {{b}} {{z}}}}}}& 0\\ 0& \frac{{{0.16666666666667}} {{{b}^{4}}}}{{1} + {{{16}} {{{b}^{4}}} {{{z}^{4}}}}{-{{{32}} {{{b}^{3}}} {{{z}^{3}}}}} + {{{24}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{8}} {{b}} {{z}}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& -{\frac{{{0.083333333333333}} {{{b}^{4}}}}{{1} + {{{16}} {{{b}^{4}}} {{{z}^{4}}}}{-{{{32}} {{{b}^{3}}} {{{z}^{3}}}}} + {{{24}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{8}} {{b}} {{z}}}}}}\\ 0& 0& \frac{{{0.083333333333333}} {{{b}^{4}}}}{{1} + {{{16}} {{{b}^{4}}} {{{z}^{4}}}}{-{{{32}} {{{b}^{3}}} {{{z}^{3}}}}} + {{{24}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{8}} {{b}} {{z}}}}}& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& -{\frac{{{0.16666666666667}} {{{b}^{4}}}}{{1} + {{{16}} {{{b}^{4}}} {{{z}^{4}}}}{-{{{32}} {{{b}^{3}}} {{{z}^{3}}}}} + {{{24}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{8}} {{b}} {{z}}}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{{{0.16666666666667}} {{{b}^{4}}}}{{1} + {{{16}} {{{b}^{4}}} {{{z}^{4}}}}{-{{{32}} {{{b}^{3}}} {{{z}^{3}}}}} + {{{24}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{8}} {{b}} {{z}}}}}& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{{{0.083333333333333}} {{{b}^{4}}}}{{1} + {{{16}} {{{b}^{4}}} {{{z}^{4}}}}{-{{{32}} {{{b}^{3}}} {{{z}^{3}}}}} + {{{24}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{8}} {{b}} {{z}}}}}\\ 0& 0& 0& 0\\ 0& -{\frac{{{0.083333333333333}} {{{b}^{4}}}}{{1} + {{{16}} {{{b}^{4}}} {{{z}^{4}}}}{-{{{32}} {{{b}^{3}}} {{{z}^{3}}}}} + {{{24}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{8}} {{b}} {{z}}}}}}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{{0.083333333333333}} {{{b}^{4}}}}{{1} + {{{16}} {{{b}^{4}}} {{{z}^{4}}}}{-{{{32}} {{{b}^{3}}} {{{z}^{3}}}}} + {{{24}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{8}} {{b}} {{z}}}}}\\ 0& 0& -{\frac{{{0.083333333333333}} {{{b}^{4}}}}{{1} + {{{16}} {{{b}^{4}}} {{{z}^{4}}}}{-{{{32}} {{{b}^{3}}} {{{z}^{3}}}}} + {{{24}} {{{b}^{2}}} {{{z}^{2}}}}{-{{{8}} {{b}} {{z}}}}}}& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$