Weyl’s theorem on complete reducibility

Theorem (Weyl complete reducibility)

Let $\mathfrak g$ be a finite-dimensional semisimple Lie algebra over a field of characteristic $0$ (in particular over $\mathbb C$). Then every finite-dimensional representation of $\mathfrak g$ is completely reducible: if $W\subseteq V$ is a subrepresentation, there exists a $\mathfrak g$-invariant subspace $W'\subseteq V$ such that

Compact Lie group analogue

If $G$ is a compact Lie group, then every finite-dimensional continuous representation of $G$ on a real or complex vector space admits a $G$-invariant inner product (obtained by averaging), hence is completely reducible. This is a key input to results like the Peter–Weyl theorem.

Context and consequences

Complete reducibility is the representation-theoretic backbone of highest-weight theory: it guarantees semisimplicity of the $\mathfrak h$-action and enables the weight space decomposition, which in turn supports the classification of irreducibles by highest weights. It also interacts with structural criteria for semisimplicity, such as nondegeneracy of the Killing form (compare equivalent characterizations of semisimplicity).