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The Killing form satisfies B([x,y],z)=B(x,[y,z]).
Let be a finite-dimensional Lie algebra over a field of characteristic , and let be its Killing form:
Lemma (ad-invariance). For all ,
Equivalently, , i.e. each is skew-adjoint with respect to .
Context. Ad-invariance is the structural feature that makes the Killing form useful in studying ideals and decompositions; it is crucial in the proof that nondegeneracy of B is equivalent to semisimplicity.
A Lie algebra is a vector space (typically over or ) equipped with a bilinear map
called the Lie bracket, such that for all :
Let be a finite-dimensional Lie algebra over a field of characteristic (typically or ). Let be the adjoint representation.
Definition (Killing form). The Killing form on is the symmetric bilinear form
It is ad-invariant and depends functorially on .
Let be a finite-dimensional Lie algebra over a field of characteristic , and let be its Killing form.
Theorem (Cartan criterion via Killing form). is semisimple if and only if the Killing form is nondegenerate.