In algebra, Weyl's theorem on complete reducibility is a fundamental result in the theory of Lie algebra representations. Let
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.
References
Weyl's theorem on complete reducibility Wikipedia(Text) CC BY-SA