Levi Decomposition Theorem

The Levi Decomposition Theorem, in Lie theory, states that every finite-dimensional Lie algebra over a field of characteristic zero can be written as a semidirect product of two pieces: its radical, the maximal solvable ideal contained in the algebra, and a semisimple subalgebra called a Levi subalgebra, complementary to the radical. The result was conjectured by Wilhelm Killing and Elie Cartan and proved by Eugenio Elia Levi, and it reduces much of the study of general finite-dimensional Lie algebras to the two more tractable cases of solvable and semisimple Lie algebras.