Weyl's theorem on complete reducibility

Remarks

One proof first treats a submodule of codimension one, using the Casimir element of a faithful quotient of $\mathfrak{g}$, which acts on a nontrivial irreducible module by a nonzero scalar. The general case then follows by applying this to a module of linear maps $\operatorname{Hom}(V, W)$. Another proof uses Whitehead's first lemma, $H^1(\mathfrak{g}, M) = 0$ for finite-dimensional $M$, with $M = \operatorname{Hom}(V, W)$. Example: the adjoint module of $\mathfrak{g}$ splits as the direct sum of the simple ideals of $\mathfrak{g}$. Semisimplicity is needed: the one-dimensional Lie algebra $F$ acting on $F^2$ by $t \mapsto \begin{pmatrix} 0 & t \\ 0 & 0 \end{pmatrix}$ preserves the line $F e_1$, and no other line, so this line has no complement. Finite dimension is needed: the Verma module $M(0)$ of $\mathfrak{sl}_2(\C)$ has a proper nonzero submodule with no complement. Characteristic $0$ is needed: over a field of characteristic $p > 0$, $\mathfrak{sl}_2$ has finite-dimensional modules that are not direct sums of irreducible ones.

Going deeper

For a student

Together with the theorem of the highest weight, it reduces the finite-dimensional representation theory of a complex semisimple Lie algebra to the study of dominant integral weights. Weyl first proved it with the unitary trick, by integration over a compact Lie group with the same complexified Lie algebra; algebraic proofs came later. It is also used to show that every representation of a semisimple Lie algebra preserves the abstract Jordan decomposition of its elements. The cohomological proof generalises: Whitehead's second lemma, $H^2(\mathfrak{g}, M) = 0$, gives the Levi decomposition.