Weyl's Theorem on Complete Reducibility

Weyl's Theorem on Complete Reducibility is a fundamental result in the representation theory of Lie algebras, named for Hermann Weyl. It states that for a semisimple Lie algebra over a field of characteristic zero, every finite-dimensional representation, or module, of that algebra decomposes as a direct sum of irreducible representations, so any such representation is completely reducible with no indecomposable pieces left over beyond the irreducibles themselves. The theorem is a cornerstone of the classical theory of Lie algebra representations, underlying the classification of finite-dimensional representations of semisimple Lie algebras.