A list of vectors S defines a basis for a Lie sub-algebra if [x, y] in span(S) 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 S0.  The program ComplementaryBasis is then used to calculate a complement C to S0.  The list of vectors B = [S, C] then gives a basis for the entire Lie algebra g.  For each x, y in S, the bracket [x, y] 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(...).