A list of vectors $S$ in a Lie algebra $\mathrm{\𝔤}\mathit{ }$is a basis for an ideal in $\mathrm{\𝔤}\mathit{ }$ if $\left[x\, y\right] \in$span($S\)$for all $x \in S$and $y \in \mathrm{\𝔤}$.

Query(S, "Ideal") returns true if the subalgebra S defines an ideal and false otherwise.

Query(S, parm, "Ideal") returns a sequence TF, Eq, Soln, IdealList.  Here TF is true if Maple finds parameter values for which S is an ideal and false otherwise; Eq is the set of equations (with the variables parm as unknowns) which must be satisfied for S to be an ideal; Soln is the list of solutions to the equations Eq; and IdealList is the list of ideals obtained from the parameter values given by the different solutions in Soln.

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)\:$

$L≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{Alg}\,\left[4\right]\right]\,\left[\left[\left[1\,4\,1\right]\,0\right]\,\left[\left[2\,3\,1\right]\,1\right]\,\left[\left[2\,4\,2\right]\,1\right]\,\left[\left[3\,4\,3\right]\,-1\right]\right]\right]\right)$

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

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

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

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

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

$\mathrm{true}$

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

$\mathrm{S2}:=\left[\mathrm{e3}\,\mathrm{e4}\right]$

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

$\mathrm{false}$

$\mathrm{S3}≔\mathrm{evalDG}\left(\left[\mathrm{e2}\,\mathrm{e1}+\mathrm{a1}\mathrm{e4}\right]\right)\:$

$\mathrm{TF},\mathrm{EQ},\mathrm{SOLN},\mathrm{IdealList}≔\mathrm{Query}\left(\mathrm{S3}\,\left\{\mathrm{a1}\,\mathrm{a2}\right\}\,''Ideal''\right)$

$\mathrm{TF}\,\mathrm{EQ}\,\mathrm{SOLN}\,\mathrm{IdealList}≔\mathrm{true}\,\left\{0\,-\mathrm{a1}\,\mathrm{a1}\right\}\,\left[\left\{\mathrm{a1}\=0\,\mathrm{a2}\=\mathrm{a2}\right\}\right]\,\left[\left[\mathrm{e2}\,\mathrm{e1}\right]\right]$

$\mathrm{EQ}$

$\left\{0\,-\mathrm{a1}\,\mathrm{a1}\right\}$

$\mathrm{S4}≔\mathrm{IdealList}\left[1\right]$

$\mathrm{S4}≔\left[\mathrm{e2}\,\mathrm{e1}\right]$

$\mathrm{Query}\left(\mathrm{S4}\,''Ideal''\right)$

$\mathrm{true}$