The subspace spanned by M defines a symmetric complement to the subalgebra spanned by S if M is a reductive complement to S and [x, y] in span(S) for all x in M and all y in M.

If S, M are a symmetric pair then they are always a naturally reductive pair for any inner product B on M.

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(...).