Semisimple decompositions of Lie algebras and prehomogeneous modules

We study {\em disemisimple} Lie algebras, i.e., Lie algebras which can be written as a vector space sum of two semisimple subalgebras. We show that a Lie algebra \(\mathfrak{g}\) is disemisimple if and only if its solvable radical coincides with its nilradical and is a prehomogeneous \(\mathfrak{s}\)-module for a Levi subalgebra \(\mathfrak{s}\) of \(\mathfrak{g}\). We use the classification of prehomogeneous \(\mathfrak{s}\)-modules for simple Lie algebras \(\mathfrak{s}\) given by Vinberg to show that the solvable radical of a disemisimple Lie algebra with simple Levi subalgebra is abelian. We extend this result to disemisimple Lie algebras having no simple quotients of type \(A\).