Every Lie algebra g admits a decomposition into the semi-direct sum g = r + s, where r is the radical of g and s is a semi-simple subalgebra.  Such a decomposition is called a Levi decomposition.  Since the radical is an ideal we have [r, r] in r, [r, s] in r, and [s, s] in s.  The radical r is uniquely defined but the semi-simple subalgebra s is not.

LeviDecomposition(Alg) calculates a Levi decomposition of the Lie algebra Alg.  If no argument is given, a Levi decomposition of the current algebra is computed.

With saveTemporaryAlgebras = true, the Lie algebras created by the program as part of the Levi decomposition algorithm are not erased and can be initialized for further analysis.  Use this option in conjunction with userlevel[LeviDecomposition] := 2.  The default is saveTemporaryAlgebras = false.

A list [rad, ss] of  two lists of vectors is returned, where rad is a basis for the radical and ss is a basis for the semi-simple part.  Either list may be empty.

The Levi decomposition is computed by the algorithm presented in the papers by Patera, Winternitz and Zassenhaus, Journal of Linear Algebra and its Applications 109:197--246(1988) and D. Rand

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