The first calling sequence BracketOfSubspaces(S1, S2) calculates the span of the set of vectors [x, y] for all x in S1 and all y in S2.

The second calling sequence BracketOfSubspaces(M1, M2) calculates the span of all matrices  [a, b] = a.b - b.a  for all a in S1 and all b in S2.

A list of linearly independent vectors defining a basis for [S1, S2]  or [M1, M2] is returned.  If [S1, S2] is trivial (that is, all the vectors in S1 commute with all the vectors in S2 then an empty list is returned.

The command BracketOfSubspaces is part of the DifferentialGeometry:-LieAlgebras package.  It can be used in the form BracketOfSubspaces(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-BracketOfSubspaces(...).