Let $\mathrm{\𝔤}\mathit{ }$be a Lie algebra,$S \subset \mathrm{\𝔤}\mathit{ }$a subalgebra, and $M \subset \mathrm{\𝔤}$ a subspace. The subalgebra, subspace pair $S\, M$ is a symmetric pair if [i] $\mathrm{\𝔤}\= S \oplus M\,$[ii] $\left[x\, y\right] \in M$for $x\in S$and $y \in M \,$and [iii] $\left[x\, y\right] \in S$ for $x\in M$and $y \in M$. Note that [i] and [ii] imply that $S\,M$define a  reductive pair. If  $S\,M$is a symmetric pair then  $S\,M$is a naturally reductive pair for any inner product on $M\.$If  $S\,M$is a symmetric pair, then $M$ is called a symmetric complement to the subalgebra $S$.

Query(M, S, "SymmetricPair") returns true if the subspace M defines a symmetric complement to the subalgebra S, and false otherwise.

Query(M, S, parm, "SymmetricPair") returns a sequence TF, Eq, Soln, symmetricList. Here TF is true if Maple finds parameter values for which S is a symmetric complement and false otherwise; Eq is the set of equations (with the variables parm as unknowns) which must be satisfied for S to be a symmetric complement; Soln is the list of solutions to the equations Eq; and symmetricList is the list of symmetric complements 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{e4}\right]\:$$\mathrm{M1}≔\left[\mathrm{e2}\,\mathrm{e3}\right]\:$

$\mathrm{Query}\left(\mathrm{S1}\,\mathrm{M1}\,''SymmetricPair''\right)$

$\mathrm{true}$

$\mathrm{S2}≔\left[\mathrm{e1}\,\mathrm{e4}\right]\:$$\mathrm{M2}≔\left[\mathrm{e2}+\mathrm{a1}\mathrm{e1}+\mathrm{a2}\mathrm{e4}\,\mathrm{e3}+\mathrm{a3}\mathrm{e1}+\mathrm{a4}\mathrm{e4}\right]\:$

$\mathrm{TF},\mathrm{EQ},\mathrm{SOLN},\mathrm{symPair}≔\mathrm{Query}\left(\mathrm{S2}\,\mathrm{M2}\,\left\{\mathrm{a1}\,\mathrm{a2}\,\mathrm{a3}\,\mathrm{a4}\right\}\,''ReductivePair''\right)$

$\mathrm{TF}\,\mathrm{EQ}\,\mathrm{SOLN}\,\mathrm{symPair}:=\mathrm{true}\,\left\{0\,\mathrm{a3}\,\mathrm{a4}\,-\mathrm{a1}\,-\mathrm{a2}\right\}\,\left[\left\{\mathrm{a1}\=0\,\mathrm{a2}\=0\,\mathrm{a3}\=0\,\mathrm{a4}\=0\right\}\right]\,\left[\left[\left[\mathrm{e1}\,\mathrm{e4}\right]\,\left[\mathrm{e2}\,\mathrm{e3}\right]\right]\right]$

$\mathrm{SOLN}$

$\left[\left\{\mathrm{a1}\=0\,\mathrm{a2}\=0\,\mathrm{a3}\=0\,\mathrm{a4}\=0\right\}\right]$

$\mathrm{S3}≔\left[\mathrm{e4}\right]\:$

$\mathrm{M3}≔\mathrm{evalDG}\left(\left[\mathrm{e1}+\mathrm{a1}\mathrm{e4}\,\mathrm{e2}\,\mathrm{e3}\right]\right)\:$

$\mathrm{Query}\left(\mathrm{S3}\,\mathrm{M3}\,''ReductivePair''\right)$

$\mathrm{true}$

$\mathrm{Query}\left(\mathrm{S3}\,\mathrm{M3}\,\left\{\mathrm{a1}\right\}\,''SymmetricPair''\right)$

$\mathrm{false}\,\left\{0\,1\,\mathrm{a1}\,-\mathrm{a1}\right\}$