A list of vectors $S$ defines a basis for a Lie subalgebra if  $\left[x\, y\right] \in$span$\left(S\right)$for all $x\,y \in S\.$

Query(S, "Subalgebra") returns true if the set S defines a subalgebra.

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

The program calculates the defining equations Eq for S to be a subalgebra as follows.  First the list of vectors $S$ is evaluated with the parameters set to zero to obtain a set of vectors $S_0$.  The program ComplementaryBasis is then used to calculate a complement $C$to $S_0$  The list of vectors $B\= \left[S\, C\right]$ then gives a basis for the entire Lie algebra $\mathrm{\𝔤}$.  For each $x\,y \in S$  the bracket $\left[x\,y\right]$is calculated and expressed as a linear combination of the vectors in the basis $B$. The components of $\[x\,y\[$ in $C$ must all vanish for $S$to be a Lie subalgebra.

We remark that the equations Eq, which the parameters must satisfy in order for S to be a subalgebra, will in general be a system of coupled quadratic equations.  Maple may not be able to solve these equations or may not solve them in full generality.

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)\:$

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

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

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

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

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

$\mathrm{false}$

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

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

$\mathrm{true}$

$\mathrm{S3}≔\mathrm{evalDG}\left(\left[\mathrm{e2}\,\mathrm{e1}+\mathrm{a1}\mathrm{e3}+\mathrm{a2}\mathrm{e4}\right]\right)\:$

$\mathrm{Query}\left(\mathrm{S3}\,\left\{\mathrm{a1}\,\mathrm{a2}\right\}\,''Subalgebra''\right)$

$\mathrm{true}\,\left\{0\,-{\mathrm{a1}}^2\,-\mathrm{a1}\mathrm{a2}\right\}\,\left[\left\{\mathrm{a1}\=0\,\mathrm{a2}\=\mathrm{a2}\right\}\right]\,\left[\left[\mathrm{e2}\,\mathrm{a2}\mathrm{e4}\+\mathrm{e1}\right]\right]$

$\mathrm{S4}≔\mathrm{evalDG}\left(\left[\mathrm{e2}\,\mathrm{e3}+\mathrm{a1}\mathrm{e2}+\mathrm{a2}\mathrm{e4}\right]\right)\:$

$\mathrm{Query}\left(\mathrm{S4}\,\left\{\mathrm{a1}\,\mathrm{a2}\right\}\,''Subalgebra''\right)$

$\mathrm{false}\,\left\{0\,1\right\}$