The Bel-Robinson tensor $B_{\mathrm{ijhk}}$ is a covariant rank 4 tensor defined in terms of the Weyl tensor $W_{\mathrm{ijhk}}$ on a 4-dimensional manifold by (see, for example, Penrose and Rindler Vol. 1)

The keyword argument indexlist = ind allows the user to specify the index structure for the Bel-Robinson tensor. For example, with indexlist = ["con", "con", "con", "con"], the contravariant form $B^{\mathrm{ijhk}}$ is returned. The default output is the purely covariant form (as above).

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form BelRobinson(...) only after executing the commands with(DifferentialGeometry); with(Tensor); in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-BelRobinson.

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{Tensor}\right)\:$

$\mathrm{DGsetup}\left(\left[x\,y\,z\,u\right]\,M\right)$

$\mathrm{frame\; name:\; M}$

$g≔\mathrm{evalDG}\left({\ⅇ}^z\mathrm{dx}\&t\mathrm{dx}\+{\ⅇ}^{-2z}\left(\mathrm{dy}\&t\mathrm{dy}\+\mathrm{dx}\&s\mathrm{du}\right)-\frac{3\mathrm{dz}\&t\mathrm{dz}}{\mathrm{\Λ}}\right)$

$g:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,{\ⅇ}^z\right]\,\left[\left[1\,4\right]\,\frac{{\ⅇ}^{-2z}}{2}\right]\,\left[\left[2\,2\right]\,{\ⅇ}^{-2z}\right]\,\left[\left[3\,3\right]\,-\frac{3}{\mathrm{\Lambda }}\right]\,\left[\left[4\,1\right]\,\frac{{\ⅇ}^{-2z}}{2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,{\ⅇ}^z\right]\,\left[\left[1\,4\right]\,\frac{{\ⅇ}^{-2z}}{2}\right]\,\left[\left[2\,2\right]\,{\ⅇ}^{-2z}\right]\,\left[\left[3\,3\right]\,-\frac{3}{\mathrm{\Lambda }}\right]\,\left[\left[4\,1\right]\,\frac{{\ⅇ}^{-2z}}{2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,{\ⅇ}^z\right]\,\left[\left[1\,4\right]\,\frac{{\ⅇ}^{-2z}}{2}\right]\,\left[\left[2\,2\right]\,{\ⅇ}^{-2z}\right]\,\left[\left[3\,3\right]\,-\frac{3}{\mathrm{\Lambda }}\right]\,\left[\left[4\,1\right]\,\frac{{\ⅇ}^{-2z}}{2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,{\ⅇ}^z\right]\,\left[\left[1\,4\right]\,\frac{{\ⅇ}^{-2z}}{2}\right]\,\left[\left[2\,2\right]\,{\ⅇ}^{-2z}\right]\,\left[\left[3\,3\right]\,-\frac{3}{\mathrm{\Lambda }}\right]\,\left[\left[4\,1\right]\,\frac{{\ⅇ}^{-2z}}{2}\right]\right]\right]\right)$

$B≔\mathrm{BelRobinson}\left(g\right)$

$B:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\,1\,1\right]\,\frac{{\mathrm{\Lambda }}^2{\ⅇ}^{2z}}{4}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\,1\,1\right]\,\frac{{\mathrm{\Lambda }}^2{\ⅇ}^{2z}}{4}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\,1\,1\right]\,\frac{{\mathrm{\Lambda }}^2{\ⅇ}^{2z}}{4}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\,1\,1\right]\,\frac{{\mathrm{\Lambda }}^2{\ⅇ}^{2z}}{4}\right]\right]\right]\right)$

$\mathrm{B1}≔\mathrm{BelRobinson}\left(g\,\mathrm{indexlist}\=\left[''con''\,''con''\,''con''\,''con''\right]\right)$

$\mathrm{B1}:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''con\_bas''\,''con\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[4\,4\,4\,4\right]\,4{\ⅇ}^{10z}{\mathrm{\Lambda }}^2\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''con\_bas''\,''con\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[4\,4\,4\,4\right]\,4{\ⅇ}^{10z}{\mathrm{\Lambda }}^2\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''con\_bas''\,''con\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[4\,4\,4\,4\right]\,4{\ⅇ}^{10z}{\mathrm{\Lambda }}^2\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''con\_bas''\,''con\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[4\,4\,4\,4\right]\,4{\ⅇ}^{10z}{\mathrm{\Lambda }}^2\right]\right]\right]\right)$

$h≔\mathrm{InverseMetric}\left(g\right)$

$h:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,4\right]\,2{\ⅇ}^{2z}\right]\,\left[\left[2\,2\right]\,{\ⅇ}^{2z}\right]\,\left[\left[3\,3\right]\,-\frac{\mathrm{\Lambda }}{3}\right]\,\left[\left[4\,1\right]\,2{\ⅇ}^{2z}\right]\,\left[\left[4\,4\right]\,-4{\ⅇ}^{5z}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,4\right]\,2{\ⅇ}^{2z}\right]\,\left[\left[2\,2\right]\,{\ⅇ}^{2z}\right]\,\left[\left[3\,3\right]\,-\frac{\mathrm{\Lambda }}{3}\right]\,\left[\left[4\,1\right]\,2{\ⅇ}^{2z}\right]\,\left[\left[4\,4\right]\,-4{\ⅇ}^{5z}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,4\right]\,2{\ⅇ}^{2z}\right]\,\left[\left[2\,2\right]\,{\ⅇ}^{2z}\right]\,\left[\left[3\,3\right]\,-\frac{\mathrm{\Lambda }}{3}\right]\,\left[\left[4\,1\right]\,2{\ⅇ}^{2z}\right]\,\left[\left[4\,4\right]\,-4{\ⅇ}^{5z}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,4\right]\,2{\ⅇ}^{2z}\right]\,\left[\left[2\,2\right]\,{\ⅇ}^{2z}\right]\,\left[\left[3\,3\right]\,-\frac{\mathrm{\Lambda }}{3}\right]\,\left[\left[4\,1\right]\,2{\ⅇ}^{2z}\right]\,\left[\left[4\,4\right]\,-4{\ⅇ}^{5z}\right]\right]\right]\right)$

$\mathrm{ContractIndices}\left(h\,B\,\left[\left[1\,1\right]\,\left[2\,2\right]\right]\right)$

$\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,0\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,0\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,0\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,0\right]\right]\right]\right)$

$C≔\mathrm{Christoffel}\left(g\right)$

$C:=\mathrm{\_DG}\left(\left[\left[''connection''\,M\,\left[\left[''con\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\,3\right]\,\mathrm{-1}\right]\,\left[\left[1\,3\,1\right]\,\mathrm{-1}\right]\,\left[\left[2\,2\,3\right]\,\mathrm{-1}\right]\,\left[\left[2\,3\,2\right]\,\mathrm{-1}\right]\,\left[\left[3\,1\,1\right]\,\frac{\mathrm{\Lambda }{\ⅇ}^z}{6}\right]\,\left[\left[3\,1\,4\right]\,-\frac{\mathrm{\Lambda }{\ⅇ}^{-2z}}{6}\right]\,\left[\left[3\,2\,2\right]\,-\frac{\mathrm{\Lambda }{\ⅇ}^{-2z}}{3}\right]\,\left[\left[3\,4\,1\right]\,-\frac{\mathrm{\Lambda }{\ⅇ}^{-2z}}{6}\right]\,\left[\left[4\,1\,3\right]\,3{\ⅇ}^{3z}\right]\,\left[\left[4\,3\,1\right]\,3{\ⅇ}^{3z}\right]\,\left[\left[4\,3\,4\right]\,\mathrm{-1}\right]\,\left[\left[4\,4\,3\right]\,\mathrm{-1}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''connection''\,M\,\left[\left[''con\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\,3\right]\,\mathrm{-1}\right]\,\left[\left[1\,3\,1\right]\,\mathrm{-1}\right]\,\left[\left[2\,2\,3\right]\,\mathrm{-1}\right]\,\left[\left[2\,3\,2\right]\,\mathrm{-1}\right]\,\left[\left[3\,1\,1\right]\,\frac{\mathrm{\Lambda }{\ⅇ}^z}{6}\right]\,\left[\left[3\,1\,4\right]\,-\frac{\mathrm{\Lambda }{\ⅇ}^{-2z}}{6}\right]\,\left[\left[3\,2\,2\right]\,-\frac{\mathrm{\Lambda }{\ⅇ}^{-2z}}{3}\right]\,\left[\left[3\,4\,1\right]\,-\frac{\mathrm{\Lambda }{\ⅇ}^{-2z}}{6}\right]\,\left[\left[4\,1\,3\right]\,3{\ⅇ}^{3z}\right]\,\left[\left[4\,3\,1\right]\,3{\ⅇ}^{3z}\right]\,\left[\left[4\,3\,4\right]\,\mathrm{-1}\right]\,\left[\left[4\,4\,3\right]\,\mathrm{-1}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''connection''\,M\,\left[\left[''con\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\,3\right]\,\mathrm{-1}\right]\,\left[\left[1\,3\,1\right]\,\mathrm{-1}\right]\,\left[\left[2\,2\,3\right]\,\mathrm{-1}\right]\,\left[\left[2\,3\,2\right]\,\mathrm{-1}\right]\,\left[\left[3\,1\,1\right]\,\frac{\mathrm{\Lambda }{\ⅇ}^z}{6}\right]\,\left[\left[3\,1\,4\right]\,-\frac{\mathrm{\Lambda }{\ⅇ}^{-2z}}{6}\right]\,\left[\left[3\,2\,2\right]\,-\frac{\mathrm{\Lambda }{\ⅇ}^{-2z}}{3}\right]\,\left[\left[3\,4\,1\right]\,-\frac{\mathrm{\Lambda }{\ⅇ}^{-2z}}{6}\right]\,\left[\left[4\,1\,3\right]\,3{\ⅇ}^{3z}\right]\,\left[\left[4\,3\,1\right]\,3{\ⅇ}^{3z}\right]\,\left[\left[4\,3\,4\right]\,\mathrm{-1}\right]\,\left[\left[4\,4\,3\right]\,\mathrm{-1}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''connection''\,M\,\left[\left[''con\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\,3\right]\,\mathrm{-1}\right]\,\left[\left[1\,3\,1\right]\,\mathrm{-1}\right]\,\left[\left[2\,2\,3\right]\,\mathrm{-1}\right]\,\left[\left[2\,3\,2\right]\,\mathrm{-1}\right]\,\left[\left[3\,1\,1\right]\,\frac{\mathrm{\Lambda }{\ⅇ}^z}{6}\right]\,\left[\left[3\,1\,4\right]\,-\frac{\mathrm{\Lambda }{\ⅇ}^{-2z}}{6}\right]\,\left[\left[3\,2\,2\right]\,-\frac{\mathrm{\Lambda }{\ⅇ}^{-2z}}{3}\right]\,\left[\left[3\,4\,1\right]\,-\frac{\mathrm{\Lambda }{\ⅇ}^{-2z}}{6}\right]\,\left[\left[4\,1\,3\right]\,3{\ⅇ}^{3z}\right]\,\left[\left[4\,3\,1\right]\,3{\ⅇ}^{3z}\right]\,\left[\left[4\,3\,4\right]\,\mathrm{-1}\right]\,\left[\left[4\,4\,3\right]\,\mathrm{-1}\right]\right]\right]\right)$

$\mathrm{nablaB1}≔\mathrm{CovariantDerivative}\left(\mathrm{B1}\,C\right)$

$\mathrm{nablaB1}:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''con\_bas''\,''con\_bas''\,''con\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[3\,4\,4\,4\,1\right]\,-\frac{2{\mathrm{\Lambda }}^3{\ⅇ}^{8z}}{3}\right]\,\left[\left[4\,3\,4\,4\,1\right]\,-\frac{2{\mathrm{\Lambda }}^3{\ⅇ}^{8z}}{3}\right]\,\left[\left[4\,4\,3\,4\,1\right]\,-\frac{2{\mathrm{\Lambda }}^3{\ⅇ}^{8z}}{3}\right]\,\left[\left[4\,4\,4\,3\,1\right]\,-\frac{2{\mathrm{\Lambda }}^3{\ⅇ}^{8z}}{3}\right]\,\left[\left[4\,4\,4\,4\,3\right]\,24{\ⅇ}^{10z}{\mathrm{\Lambda }}^2\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''con\_bas''\,''con\_bas''\,''con\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[3\,4\,4\,4\,1\right]\,-\frac{2{\mathrm{\Lambda }}^3{\ⅇ}^{8z}}{3}\right]\,\left[\left[4\,3\,4\,4\,1\right]\,-\frac{2{\mathrm{\Lambda }}^3{\ⅇ}^{8z}}{3}\right]\,\left[\left[4\,4\,3\,4\,1\right]\,-\frac{2{\mathrm{\Lambda }}^3{\ⅇ}^{8z}}{3}\right]\,\left[\left[4\,4\,4\,3\,1\right]\,-\frac{2{\mathrm{\Lambda }}^3{\ⅇ}^{8z}}{3}\right]\,\left[\left[4\,4\,4\,4\,3\right]\,24{\ⅇ}^{10z}{\mathrm{\Lambda }}^2\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''con\_bas''\,''con\_bas''\,''con\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[3\,4\,4\,4\,1\right]\,-\frac{2{\mathrm{\Lambda }}^3{\ⅇ}^{8z}}{3}\right]\,\left[\left[4\,3\,4\,4\,1\right]\,-\frac{2{\mathrm{\Lambda }}^3{\ⅇ}^{8z}}{3}\right]\,\left[\left[4\,4\,3\,4\,1\right]\,-\frac{2{\mathrm{\Lambda }}^3{\ⅇ}^{8z}}{3}\right]\,\left[\left[4\,4\,4\,3\,1\right]\,-\frac{2{\mathrm{\Lambda }}^3{\ⅇ}^{8z}}{3}\right]\,\left[\left[4\,4\,4\,4\,3\right]\,24{\ⅇ}^{10z}{\mathrm{\Lambda }}^2\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''con\_bas''\,''con\_bas''\,''con\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[3\,4\,4\,4\,1\right]\,-\frac{2{\mathrm{\Lambda }}^3{\ⅇ}^{8z}}{3}\right]\,\left[\left[4\,3\,4\,4\,1\right]\,-\frac{2{\mathrm{\Lambda }}^3{\ⅇ}^{8z}}{3}\right]\,\left[\left[4\,4\,3\,4\,1\right]\,-\frac{2{\mathrm{\Lambda }}^3{\ⅇ}^{8z}}{3}\right]\,\left[\left[4\,4\,4\,3\,1\right]\,-\frac{2{\mathrm{\Lambda }}^3{\ⅇ}^{8z}}{3}\right]\,\left[\left[4\,4\,4\,4\,3\right]\,24{\ⅇ}^{10z}{\mathrm{\Lambda }}^2\right]\right]\right]\right)$

$\mathrm{Divergence}≔\mathrm{ContractIndices}\left(\mathrm{nablaB1}\,\left[\left[1\,5\right]\right]\right)$

$\mathrm{Divergence}:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''con\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\,1\right]\,0\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''con\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\,1\right]\,0\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''con\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\,1\right]\,0\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''con\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\,1\right]\,0\right]\right]\right]\right)$

$R≔\mathrm{RicciTensor}\left(g\right)$

$R:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,\mathrm{\Lambda }{\ⅇ}^z\right]\,\left[\left[1\,4\right]\,\frac{\mathrm{\Lambda }{\ⅇ}^{-2z}}{2}\right]\,\left[\left[2\,2\right]\,\mathrm{\Lambda }{\ⅇ}^{-2z}\right]\,\left[\left[3\,3\right]\,\mathrm{-3}\right]\,\left[\left[4\,1\right]\,\frac{\mathrm{\Lambda }{\ⅇ}^{-2z}}{2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,\mathrm{\Lambda }{\ⅇ}^z\right]\,\left[\left[1\,4\right]\,\frac{\mathrm{\Lambda }{\ⅇ}^{-2z}}{2}\right]\,\left[\left[2\,2\right]\,\mathrm{\Lambda }{\ⅇ}^{-2z}\right]\,\left[\left[3\,3\right]\,\mathrm{-3}\right]\,\left[\left[4\,1\right]\,\frac{\mathrm{\Lambda }{\ⅇ}^{-2z}}{2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,\mathrm{\Lambda }{\ⅇ}^z\right]\,\left[\left[1\,4\right]\,\frac{\mathrm{\Lambda }{\ⅇ}^{-2z}}{2}\right]\,\left[\left[2\,2\right]\,\mathrm{\Lambda }{\ⅇ}^{-2z}\right]\,\left[\left[3\,3\right]\,\mathrm{-3}\right]\,\left[\left[4\,1\right]\,\frac{\mathrm{\Lambda }{\ⅇ}^{-2z}}{2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,\mathrm{\Lambda }{\ⅇ}^z\right]\,\left[\left[1\,4\right]\,\frac{\mathrm{\Lambda }{\ⅇ}^{-2z}}{2}\right]\,\left[\left[2\,2\right]\,\mathrm{\Lambda }{\ⅇ}^{-2z}\right]\,\left[\left[3\,3\right]\,\mathrm{-3}\right]\,\left[\left[4\,1\right]\,\frac{\mathrm{\Lambda }{\ⅇ}^{-2z}}{2}\right]\right]\right]\right)$

$\mathrm{evalDG}\left(R-\mathrm{\Λ}g\right)$

$\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,0\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,0\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,0\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,0\right]\right]\right]\right)$

$W≔\mathrm{WeylTensor}\left(g\right)$

$W:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,2\,1\,2\right]\,-\frac{\mathrm{\Lambda }{\ⅇ}^{-z}}{2}\right]\,\left[\left[1\,2\,2\,1\right]\,\frac{\mathrm{\Lambda }{\ⅇ}^{-z}}{2}\right]\,\left[\left[1\,3\,1\,3\right]\,-\frac{3{\ⅇ}^z}{2}\right]\,\left[\left[1\,3\,3\,1\right]\,\frac{3{\ⅇ}^z}{2}\right]\,\left[\left[2\,1\,1\,2\right]\,\frac{\mathrm{\Lambda }{\ⅇ}^{-z}}{2}\right]\,\left[\left[2\,1\,2\,1\right]\,-\frac{\mathrm{\Lambda }{\ⅇ}^{-z}}{2}\right]\,\left[\left[3\,1\,1\,3\right]\,\frac{3{\ⅇ}^z}{2}\right]\,\left[\left[3\,1\,3\,1\right]\,-\frac{3{\ⅇ}^z}{2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,2\,1\,2\right]\,-\frac{\mathrm{\Lambda }{\ⅇ}^{-z}}{2}\right]\,\left[\left[1\,2\,2\,1\right]\,\frac{\mathrm{\Lambda }{\ⅇ}^{-z}}{2}\right]\,\left[\left[1\,3\,1\,3\right]\,-\frac{3{\ⅇ}^z}{2}\right]\,\left[\left[1\,3\,3\,1\right]\,\frac{3{\ⅇ}^z}{2}\right]\,\left[\left[2\,1\,1\,2\right]\,\frac{\mathrm{\Lambda }{\ⅇ}^{-z}}{2}\right]\,\left[\left[2\,1\,2\,1\right]\,-\frac{\mathrm{\Lambda }{\ⅇ}^{-z}}{2}\right]\,\left[\left[3\,1\,1\,3\right]\,\frac{3{\ⅇ}^z}{2}\right]\,\left[\left[3\,1\,3\,1\right]\,-\frac{3{\ⅇ}^z}{2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,2\,1\,2\right]\,-\frac{\mathrm{\Lambda }{\ⅇ}^{-z}}{2}\right]\,\left[\left[1\,2\,2\,1\right]\,\frac{\mathrm{\Lambda }{\ⅇ}^{-z}}{2}\right]\,\left[\left[1\,3\,1\,3\right]\,-\frac{3{\ⅇ}^z}{2}\right]\,\left[\left[1\,3\,3\,1\right]\,\frac{3{\ⅇ}^z}{2}\right]\,\left[\left[2\,1\,1\,2\right]\,\frac{\mathrm{\Lambda }{\ⅇ}^{-z}}{2}\right]\,\left[\left[2\,1\,2\,1\right]\,-\frac{\mathrm{\Lambda }{\ⅇ}^{-z}}{2}\right]\,\left[\left[3\,1\,1\,3\right]\,\frac{3{\ⅇ}^z}{2}\right]\,\left[\left[3\,1\,3\,1\right]\,-\frac{3{\ⅇ}^z}{2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,2\,1\,2\right]\,-\frac{\mathrm{\Lambda }{\ⅇ}^{-z}}{2}\right]\,\left[\left[1\,2\,2\,1\right]\,\frac{\mathrm{\Lambda }{\ⅇ}^{-z}}{2}\right]\,\left[\left[1\,3\,1\,3\right]\,-\frac{3{\ⅇ}^z}{2}\right]\,\left[\left[1\,3\,3\,1\right]\,\frac{3{\ⅇ}^z}{2}\right]\,\left[\left[2\,1\,1\,2\right]\,\frac{\mathrm{\Lambda }{\ⅇ}^{-z}}{2}\right]\,\left[\left[2\,1\,2\,1\right]\,-\frac{\mathrm{\Lambda }{\ⅇ}^{-z}}{2}\right]\,\left[\left[3\,1\,1\,3\right]\,\frac{3{\ⅇ}^z}{2}\right]\,\left[\left[3\,1\,3\,1\right]\,-\frac{3{\ⅇ}^z}{2}\right]\right]\right]\right)$

$\mathrm{BelRobinson}\left(g\,W\right)$

$\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\,1\,1\right]\,\frac{{\mathrm{\Lambda }}^2{\ⅇ}^{2z}}{4}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\,1\,1\right]\,\frac{{\mathrm{\Lambda }}^2{\ⅇ}^{2z}}{4}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\,1\,1\right]\,\frac{{\mathrm{\Lambda }}^2{\ⅇ}^{2z}}{4}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\,1\,1\right]\,\frac{{\mathrm{\Lambda }}^2{\ⅇ}^{2z}}{4}\right]\right]\right]\right)$