Compact Lie algebra is reductive

Let $G$ be a compact Lie group with Lie algebra $\mathfrak g$.

Theorem (Compact implies reductive).
The Lie algebra $\mathfrak g$ is reductive: it decomposes as a direct sum of ideals

where $Z(\mathfrak g)$ is the center (an abelian ideal, compare abelian Lie algebras ) and $[\mathfrak g,\mathfrak g]$ is semisimple (see semisimple Lie algebra ). In particular, $[\mathfrak g,\mathfrak g]$ has nondegenerate Killing form, while the Killing form of $\mathfrak g$ is negative semidefinite with kernel $Z(\mathfrak g)$ (compare Killing form and nondegeneracy vs semisimplicity ).

Idea of proof.
Compactness gives an $\mathrm{Ad}(G)$-invariant inner product on $\mathfrak g$ by averaging any inner product over $G$; equivalently, compact $G$ admits a bi-invariant metric . With such an inner product, $\mathrm{ad}_x$ is skew-adjoint for all $x$, forcing strong structural constraints that split off the center and make the derived ideal semisimple.

Context.
Reductivity is the Lie-algebra reflection of robust representation theory for compact groups: finite-dimensional representations are completely reducible (compare complete reducibility and Peter–Weyl theorem ).