The Killing form is an inner product on a finite dimensional Lie algebra g defined by B(X, Y) = Tr(ad(X) ad(Y)) in the adjoint representation, where ad(X) is the adjoint representation of X. (-1) is adjoint-invariant in the sense that B(ad(X) Y, Z) = - B(Y, ad(X) Z). When g is a semisimple Lie algebra, the Killing form is nondegenerate.

Cartan matrix | inner product | Killing's equation | Killing vectors | Lie algebra | matrix signature | semisimple Lie algebra | trace form | Weyl group