A subspace M defines a reductive complement to the subalgebra S in a Lie algebra g if g = S + M (vector space direct sum) and [x, y] in M for all x in S and all y in M.

Query(S, M, "ReductivePair") returns true if the subspace M defines a reductive complement to the subalgebra S.

Query(S, M, parm, "ReductivePair") returns a sequence TF, Eq, Soln, reductiveList.  Here TF is true if Maple finds parameter values for which M is a reductive complement and false otherwise; Eq is the set of equations (with the variables parm as unknowns) which must be satisfied for M to be a reductive complement; Soln is the list of solutions to the equations Eq; and reductiveList is the list of reductive subspaces 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(...).