Weyl’s theorem on complete reducibility

Weyl's theorem. Let $\mathfrak g$ be a finite-dimensional semisimple Lie algebra over a field of characteristic $0$ . Every finite-dimensional representation $V$ of $\mathfrak g$ is completely reducible: for each subrepresentation $W\subseteq V$ , there is a $\mathfrak g$ -invariant subspace $W'$ such that

V = W \oplus W'.

Compact Lie group analogue

If $G$ is a compact Lie group, then every finite-dimensional continuous real or complex representation admits a $G$ -invariant inner product, obtained by averaging. It is therefore completely reducible.

Equivalent characterizations

For a finite-dimensional representation, the invariant-complement property is equivalent to being a finite direct sum of irreducible representations.