A collection of subalgebras S1, S2, ... of a Lie algebra g defines a direct sum decomposition of g if g = S1 + S2 + ... (vector space direct sum) and [Si, Sj] = 0 for i <> j.

Query([S1, S2, ... ], "DirectSumDecomposition") returns true if the subspaces S1, S2, ... define a direct sum decomposition of the Lie algebra g.

Query(B, [d1, d2, ... ], "DirectSumDecomposition") returns true if the first d1 vectors in B, the second d2 in B, ... define a direct sum decomposition of g.

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(...).