Equivalent characterizations of semisimplicity for Lie algebras

Let $\mathfrak g$ be a finite-dimensional Lie algebra over a field of characteristic $0$ . The following are equivalent.

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

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.

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

Remarks

Nondegeneracy of the Killing form is Cartan's criterion for semisimplicity. Complete reducibility of every finite-dimensional representation is a consequence of semisimplicity by Weyl's theorem, but complete reducibility of the adjoint module alone is not an equivalent condition: an abelian Lie algebra has a completely reducible trivial adjoint module and is not semisimple.