Let $\mathrm{\𝔤}\mathit{ }$be a Lie algebra and let $S_1\subset \mathrm{\𝔤}$and $S_2 \subset g_{ }$be two subspaces (not necessarily subalgebras). Then $\left[S_1\, S_2\right]$denotes the span of all vectors of the form $\left[x\, y\right]$with $x \in S_1$and $y\mathit{ }\mathit{\in }{S}_1\.$If $S_1 \=$span {$x_1\, x_2\, ..\. \, x_p\}$and $S_2 \=$span {$y_1\, y_2\, ..\. \, y_q\} \,$then

The first calling sequence BracketOfSubspaces(S1, S2) calculates the subspace $\left[S_1\, S_2\right]\.$A list of linearly independent vectors defining a basis for $\left[S_1\, S_2\right]$ is returned. If $\left[S_1\, S_2\right] \= 0$(that is, all the vectors in $S_1$commute with all the vectors in $S_2$), then an empty list is returned.

The second calling sequence BracketOfSubspaces(M1, M2) calculates the subspace $\left[M_1\, M_2\right]$. A list of linearly independent vectors defining a basis for $\left[M_1\, M_2\right]$ is returned. If $\left[M_1\, M_2\right] \= 0$(that is, all the matrices in $M_1$in commute with all the matrices in $M_2$), then an empty list is returned.

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

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

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

$\mathrm{L1}:=\left[\left[\mathrm{e2}\,\mathrm{e4}\right]\=\mathrm{e1}\,\left[\mathrm{e3}\,\mathrm{e4}\right]\=\mathrm{e3}\right]$

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

$\mathrm{S1}≔\left[\mathrm{e1}\,\mathrm{e2}\right]\:$$\mathrm{S2}≔\left[\mathrm{e3}\,\mathrm{e4}\right]\:$

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

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

$\mathrm{S3}≔\left[\mathrm{e1}\,\mathrm{e2}\,\mathrm{e3}\right]\:$

$\mathrm{BracketOfSubspaces}\left(\mathrm{S3}\,\mathrm{S3}\right)$

$\mathrm{M1}≔\left[\mathrm{Matrix}\left(\left[\left[0\,1\,1\right]\,\left[0\,0\,0\right]\,\left[0\,0\,0\right]\right]\right)\,\mathrm{Matrix}\left(\left[\left[0\,1\,0\right]\,\left[0\,0\,0\right]\,\left[1\,0\,0\right]\right]\right)\right]$

$\mathrm{M2}≔\left[\mathrm{Matrix}\left(\left[\left[0\,0\,1\right]\,\left[0\,0\,0\right]\,\left[0\,0\,0\right]\right]\right)\,\mathrm{Matrix}\left(\left[\left[0\,0\,0\right]\,\left[0\,0\,1\right]\,\left[0\,0\,0\right]\right]\right)\,\mathrm{Matrix}\left(\left[\left[0\,0\,0\right]\,\left[0\,0\,0\right]\,\left[0\,0\,1\right]\right]\right)\right]$

$\mathrm{BracketOfSubspaces}\left(\mathrm{M1}\,\mathrm{M2}\right)$