Levi Decomposition

In this chapter, which owing to its brevity is not divided into sections, we prove Levi’s theorem on the decomposition of an arbitrary Lie algebra into a semidirect sum of a solvable ideal (radical) and a semisimple subalgebra and the theorem on the uniqueness of this decomposition due to A.I. Malcev. Levi’s theorem implies the result which concludes the classical Lie group theory—the existence of a Lie group with an arbitrary given tangent algebra. Next we will consider an analogue of Levi decomposition for algebraic groups.