Complete reducibility for compact Lie groups

We argue by induction on $n=\dim V$, assuming the assertion for all representation spaces of dimension $<n$. If $n=0$ then $V$ is the empty direct sum of irreducibles by [L3]; otherwise $V$ has a nonzero subrepresentation, and choosing among the nonzero subrepresentations one of least positive dimension gives an irreducible subrepresentation $W$, since any nonzero proper subrepresentation of $W$ would be a nonzero subrepresentation of $V$ of strictly smaller positive dimension by [L3].

By [L1] fix a $G$-invariant positive-definite Hermitian inner product on $V$ and let $W^{\perp }$ be the orthogonal complement of $W$; then $W^{\perp }$ is a subrepresentation, because for $w^{\mathrm{\prime }}\in W^{\perp }$, $w\in W$ and $g\in G$ unitarity and invariance of $W$ give $⟨\pi (g)w^{\mathrm{\prime }},w⟩=⟨w^{\mathrm{\prime }},\pi (g{)}^{-1}w⟩$ with $\pi (g{)}^{-1}w=\pi (g^{-1})w\in W$, so $⟨\pi (g)w^{\mathrm{\prime }},w⟩=0$.

Since $W\mathrm{\ne }0$, [L3] gives $\dim W^{\perp }=\dim V-\dim W<\dim V$, so the inductive hypothesis applies to the subrepresentation $W^{\perp }$ and exhibits it as a direct sum of irreducible subrepresentations; adjoining the irreducible summand $W$ gives $V=W\oplus W^{\perp }$ as a direct sum of irreducibles by [L2]. The Axiom of Choice entered only through [L1].