Semisimple decompositions of Lie algebras and prehomogeneous modules

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