Weyl complete reducibility theorem

Every finite-dimensional representation of a complex semisimple Lie algebra is a direct sum of irreducible representations. Equivalently, every invariant subspace has an invariant complement.

For a finite-dimensional continuous complex representation $R$ of a compact group, normalize Haar measure to mass one and average any positive-definite Hermitian inner product:The resulting inner product is positive-definite and invariant. If $W$ is an invariant subspace, then $W^{\perp }$ is invariant as well, since $⟨R(g)v,w{⟩}_K=⟨v,R(g^{-1})w{⟩}_K$. Thus $V=W\oplus W^{\perp }$. Restriction to a compact real form, followed by integration of a Lie-algebra representation, proves the Weyl complete reducibility theorem.

If a submodule $U$ of a Lie algebra representation $V$ admits an intertwining map $P:V\to U$ with $P{|}_U=I_U$, then $P^2=P$ and its kernel is an invariant complement. For a complex semisimple Lie algebra, the Hom representation on maps $V\to U$ whose restriction is a scalar identity reduces finding $P$ to splitting of a trivial quotient for a semisimple Lie algebra.

In a finite-dimensional short exact sequence $0\to N\to E\to \mathbb{C}\to 0$ of representations of a complex semisimple Lie algebra, the generalized zero eigenspace of the Casimir operator maps onto the trivial quotient. All its irreducible composition factors have zero Casimir eigenvalue, so they are trivial. The action is therefore strictly upper triangular and has solvable image; because a semisimple Lie algebra is a perfect Lie algebra, that image is zero. Every lift in this generalized zero eigenspace is invariant, giving a split short exact sequence.