Let $\mathrm{\𝔤}\mathit{ }$be a Lie algebra and let $h \subset k \subset \mathrm{\𝔤}\mathit{ }$be subalgebras.The normalizer $n$ of $h$ in $k$ is the largest subalgebra $n$ of $k$ which contains $h$ as an ideal. The normalizer of $h$ always contains $h$ itself.

SubalgebraNormalizer(h, k) calculates the normalizer of $h$ in the subalgebra $k$. If the second argument $k$ is not specified, then the default is $k \=\mathrm{\𝔤}\mathit{ }$and the normalizer of $h$in $\mathrm{\𝔤}\mathit{ }$is calculated.

A list of vectors defining a basis for the normalizer of $h$ is returned.

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

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

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

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

$\mathrm{MultiplicationTable}\left(''LieBracket''\right)$

$\left[\left[\mathrm{e2}\,\mathrm{e5}\right]\=\mathrm{e1}\,\left[\mathrm{e3}\,\mathrm{e4}\right]\=\mathrm{e1}\,\left[\mathrm{e3}\,\mathrm{e5}\right]\=\mathrm{e2}\right]$

$\mathrm{S1}≔\left[\mathrm{e3}\right]\:$$\mathrm{S2}≔\left[\mathrm{e1}\,\mathrm{e3}\,\mathrm{e4}\right]\:$

$\mathrm{SubalgebraNormalizer}\left(\mathrm{S1}\,\mathrm{S2}\right)$

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

$\mathrm{S3}≔\left[\mathrm{e2}\,\mathrm{e4}\right]\:$$\mathrm{S4}≔\left[\mathrm{e1}\,\mathrm{e2}\,\mathrm{e4}\,\mathrm{e5}\right]\:$

$\mathrm{SubalgebraNormalizer}\left(\mathrm{S3}\,\mathrm{S4}\right)$

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

$\mathrm{S5}≔\left[\mathrm{e1}\,\mathrm{e2}\right]\:$

$\mathrm{SubalgebraNormalizer}\left(\mathrm{S5}\right)$

$\left[\mathrm{e5}\,\mathrm{e4}\,\mathrm{e3}\,\mathrm{e2}\,\mathrm{e1}\right]$