Let $\mathrm{\𝔤}\mathit{ }$be a Lie algebra, $k$a subalgebra of $\mathrm{\𝔤}$, and $\mathrm{\𝔥}\mathit{ }$an ideal with $\mathrm{\𝔥}\mathit{ }\mathit{\subset }k\mathit{ }\mathit{\subset }\mathrm{\𝔤}\mathit{ }$. Then the generalized center $\mathrm{\𝔠}\mathit{ }$of $h$with respect to $k$is the ideal $\mathrm{\𝔠}\mathit{ }\mathit{\=}\mathit{\{}x \in k \| \left[x\, y\right] \in h$ for all $y \in k \}\.$ In particular, the generalized center of $h$in $\mathrm{\𝔤}$ is the inverse image of the center of the quotient algebra $\mathrm{\𝔤}\mathit{\/}\mathrm{\𝔥}$ with respect to the canonical projection map $\mathrm{\π}\:\mathit{ }\mathrm{\𝔤}\mathit{ }\mathit{\to }\mathit{ }$$\mathrm{\𝔤}\mathit{\/}\mathrm{\𝔥}$.

A list of vectors defining a basis for the generalized center of $\mathrm{\𝔥}\mathit{ }$in $k$is returned. If the optional argument S2 is omitted, then the default is $k \= \mathrm{\𝔤}\.$If the generalized center of $\mathrm{\𝔥}$ in $k$is trivial, then an empty list is returned.

The command GeneralizedCenter is part of the DifferentialGeometry:-LieAlgebras package.  It can be used in the form GeneralizedCenter(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-GeneralizedCenter(...).

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

$\mathrm{L1}≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{Alg1}\,\left[5\right]\right]\,\left[\left[\left[2\,5\,1\right]\,1\right]\,\left[\left[3\,5\,2\right]\,1\right]\,\left[\left[4\,5\,4\right]\,1\right]\right]\right]\right)\:$

$\mathrm{DGsetup}\left(\mathrm{L1}\right)\:$

$\mathrm{S1}≔\left[\mathrm{e1}\,\mathrm{e2}\right]\:$

$\mathrm{GeneralizedCenter}\left(\left[\mathrm{e1}\,\mathrm{e2}\right]\right)$

$\left[\mathrm{e3}\,\mathrm{e2}\,\mathrm{e1}\right]$

$\mathrm{S2}≔\left[\mathrm{e1}\,\mathrm{e4}\right]\:$$\mathrm{S3}≔\left[\mathrm{e1}\,\mathrm{e2}\,\mathrm{e4}\,\mathrm{e5}\right]\:$

$\mathrm{GeneralizedCenter}\left(\mathrm{S2}\,\mathrm{S3}\right)$

$\left[\mathrm{e5}\,\mathrm{e1}\,\mathrm{e2}\,\mathrm{e4}\right]$