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. Let g be a semisimple Lie algebra over a field of characteristic zero. The theorem states that every finite-dimensional module over g is semisimple as a module (i.e., a direct sum of simple modules.)

The theorem is a consequence of Whitehead's lemma (see Weibel's homological algebra book). Weyl's original proof was analytic in nature: it famously used the unitarian trick. The usual algebraic proof makes use of the Casimir element of the universal enveloping algebra.