Weyl's theorem on complete reducibility

In algebra, Weyl's theorem on complete reducibility is a fundamental result in the theory of Lie algebra representations (specifically in the representation theory of semisimple Lie algebras). Let ${\displaystyle {\mathfrak {g}}}$ be a semisimple Lie algebra over a field of characteristic zero. The theorem states that every finite-dimensional module over ${\displaystyle {\mathfrak {g}}}$ is semisimple as a module (i.e., a direct sum of simple modules.)^[1]