Let $\mathrm{\𝔤}\mathit{ }$be a Lie algebra, $S \subset \mathrm{\𝔤}$ a subalgebra, and $M \subset \mathrm{\𝔤}$ a subspace. Let $B$ be a non-degenerate inner product on $M$.  Then the subalgebra, subspace pair $S\, M$ is called naturally reductive with respect to the inner product $B$if [i] the subspace $M$ defines a reductive complement to the subalgebra $S\,$ and [ii] the inner product $B$ is $\mathrm{\𝔤}\mathit{ }$ invariant, that is, $B\left({\left[x\,y\right]}_M \,z\right) \+ B\left(y\, {\left[x\,z\right]}_M\right) \= 0$for all $x \in \mathrm{\𝔤}$ and $y\,z \in M$.  Here ${\left[x\,y\right]}_M$ denotes the $M-$component of $\left[x\,y\right]$with respect to the decomposition $\mathrm{\𝔤}\= S \oplus M$.

Query(S, M, B, "NaturallyReductivePair") returns true if S, M is naturally reductive with respect to the inner product B, and false otherwise.

Query(S, M, B, parm, "NaturallyReductivePair") returns a sequence TF, Eq, Soln, NatRedPair.  Here TF is true if Maple finds parameter values for which the pair S, M is naturally reductive and false otherwise; Eq is the set of equations (with the variables parm as unknowns) which must be satisfied for S, M to be naturally reductive; Soln is the list of solutions to the equations Eq; and NatRedPair is the list of naturally reductive subspaces and inner products 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)\:$

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

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

$\mathrm{S1}≔\left[\mathrm{e2}\right]\:$$\mathrm{M1}≔\left[\mathrm{e1}\,\mathrm{e3}\right]\:$$\mathrm{B1}≔\mathrm{Matrix}\left(\left[\left[0\,1\right]\,\left[1\,0\right]\right]\right)$

$\mathrm{Query}\left(\mathrm{S1}\,\mathrm{M1}\,\mathrm{B1}\,''NaturallyReductivePair''\right)$

$\mathrm{true}$

$g≔\mathrm{evalDG}\left(\mathrm{\θ1}\&t\mathrm{\θ3}+\mathrm{\θ3}\&t\mathrm{\θ1}\right)$

$g:=\mathrm{\θ1}\mathrm{\θ3}\+\mathrm{\θ3}\mathrm{\θ1}$

$\mathrm{LieDerivative}\left(\mathrm{e2}\,g\right)$

$0\mathrm{\θ1}\mathrm{\θ1}$

$\mathrm{LieDerivative}\left(\mathrm{e1}\,g\right)$

$-\mathrm{\θ2}\mathrm{\θ3}-\mathrm{\θ3}\mathrm{\θ2}$

$\mathrm{LieDerivative}\left(\mathrm{e3}\,g\right)$

$\mathrm{\θ1}\mathrm{\θ2}\+\mathrm{\θ2}\mathrm{\θ1}$

$\mathrm{L2}≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{Alg2}\,\left[4\right]\right]\,\left[\left[\left[1\,4\,1\right]\,1\right]\,\left[\left[2\,4\,2\right]\,b\right]\,\left[\left[3\,4\,2\right]\,1\right]\,\left[\left[2\,4\,3\right]\,-1\right]\,\left[\left[3\,4\,3\right]\,b\right]\right]\right]\right)$

$\mathrm{L2}:=\left[\left[\mathrm{e1}\,\mathrm{e4}\right]\=\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e4}\right]\=b\mathrm{e2}-\mathrm{e3}\,\left[\mathrm{e3}\,\mathrm{e4}\right]\=\mathrm{e2}\+b\mathrm{e3}\right]$

$\mathrm{DGsetup}\left(\mathrm{L2}\right)\:$

$\mathrm{S2}≔\left[\mathrm{e1}\,\mathrm{e4}\right]\:$$\mathrm{M2}≔\left[\mathrm{e2}\,\mathrm{e3}\right]\:$$\mathrm{B2}≔\mathrm{Matrix}\left(\left[\left[r\,s\right]\,\left[s\,t\right]\right]\right)$

$\mathrm{Query}\left(\mathrm{S2}\,\mathrm{M2}\,\mathrm{B2}\,''NaturallyReductivePair''\right)$

$\mathrm{false}$

$\mathrm{TF},\mathrm{EQ},\mathrm{SOLN},\mathrm{natRedPair}≔\mathrm{Query}\left(\mathrm{S2}\,\mathrm{M2}\,\mathrm{B2}\,\left\{b\,r\,s\,t\right\}\,''NaturallyReductivePair''\right)$

$\mathrm{L3}≔\mathrm{subs}\left(b=0\,\mathrm{L2}\right)$

$\mathrm{L3}:=\left[\left[\mathrm{e1}\,\mathrm{e4}\right]\=\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e4}\right]\=-\mathrm{e3}\,\left[\mathrm{e3}\,\mathrm{e4}\right]\=\mathrm{e2}\right]$

$\mathrm{DGsetup}\left(\mathrm{L3}\right)$

$\mathrm{Lie\; algebra:\; Alg2}$

$\mathrm{S3}≔\mathrm{natRedPair}\left[1\right]\left[1\right]\;$$\mathrm{M3}≔\mathrm{natRedPair}\left[1\right]\left[2\right]\;$$\mathrm{B3}≔\mathrm{natRedPair}\left[1\right]\left[3\right]$

$\mathrm{S3}:=\left[\mathrm{e1}\,\mathrm{e4}\right]$

$\mathrm{M3}:=\left[\mathrm{e2}\,\mathrm{e3}\right]$

$\mathrm{Query}\left(\mathrm{S3}\,\mathrm{M3}\,\mathrm{B3}\,''NaturallyReductivePair''\right)$

$\mathrm{true}$