A Lie algebra $\mathrm{\𝔤}\mathit{ }$ is Abelian if $\left[x\, y\right] \= 0$for all $x\,y \in \mathrm{\𝔤}$.  

Query(Alg, "Abelian") returns true if Alg is an Abelian Lie algebra and false otherwise.  If the algebra is unspecified, then Query is applied to the current algebra.

Query(S, "Abelian") returns true if the subalgebra S is an Abelian Lie algebra and false otherwise.

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

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

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

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

$\mathrm{L2}≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{Alg2}\,\left[3\right]\right]\,\left[\right]\right]\right)$

$\mathrm{L2}:=\left[\right]$

$\mathrm{DGsetup}\left(\mathrm{L1}\,\left[x\right]\,\left[a\right]\right)\:$$\mathrm{DGsetup}\left(\mathrm{L2}\,\left[y\right]\,\left[b\right]\right)\:$

$\mathrm{Query}\left(\mathrm{Alg1}\,''Abelian''\right)$

$\mathrm{false}$

$\mathrm{ChangeLieAlgebraTo}\left(\mathrm{Alg2}\right)\:$

$\mathrm{Query}\left(''Abelian''\right)$

$\mathrm{true}$

$\mathrm{S1}≔\left[\mathrm{x1}\,\mathrm{x2}\right]\:$$\mathrm{S2}≔\left[\mathrm{x2}\,\mathrm{x3}\right]\:$

$\mathrm{Query}\left(\mathrm{S1}\,''Abelian''\right)$

$\mathrm{false}$

$\mathrm{Query}\left(\mathrm{S2}\,''Abelian''\right)$

$\mathrm{true}$