Equivalent characterizations of semisimplicity for Lie algebras

Theorem (TFAE: semisimplicity)

Let $\mathfrak g$ be a finite-dimensional Lie algebra over a field of characteristic $0$ (in particular over $\mathbb R$ or $\mathbb C$). The following are equivalent.

1. No nonzero solvable ideals: $\mathfrak g$ is semisimple, i.e. it has no nonzero solvable ideal (equivalently, its radical is $0$).

1. Nondegenerate Killing form: the Killing form $\kappa(X,Y)=\mathrm{tr}(\mathrm{ad}_X\mathrm{ad}_Y)$ is nondegenerate; compare nondegeneracy of the Killing form.

1. Direct sum of simple ideals: $\mathfrak g$ is a (finite) direct sum of simple Lie algebras; see semisimple equals direct sum of simple ideals.

1. Adjoint representation is completely reducible: the representation $\mathrm{ad}:\mathfrak g\to \mathfrak{gl}(\mathfrak g)$ is completely reducible.

1. Cartan-type trace criterion: $\mathfrak g$ satisfies a trace criterion equivalent to semisimplicity as in Cartan’s criterion for semisimplicity.

Context

These equivalences are used interchangeably in practice: (2) is often the fastest computational test, while (3) is the structural starting point for classification (compare classification of simple Lie algebras). Complete reducibility of representations (see Weyl’s theorem) is a major consequence of semisimplicity.