Abstract
We study 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 g is disemisimple if and only if its solvable radical coincides with its nilradical and is a prehomogeneous s-module for a Levi subalgebra s of g. We use the classification of prehomogeneous s-modules for simple Lie algebras 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.
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