Killing Form

The Killing form is a symmetric bilinear form on a Lie algebra defined by the trace of the composition of two adjoint endomorphisms: for elements X and Y, the Killing form is κ(X,Y) = tr(ad_X ∘ ad_Y). Introduced by Wilhelm Killing in his study of the structure of Lie algebras, this form provides a canonical metric that measures the intrinsic curvature of the algebra's multiplication table. A Lie algebra is semisimple if and only if its Killing form is nondegenerate — a criterion known as Cartan's criterion that connects algebraic health to geometric nondegeneracy.

The Killing form is the ancestor of a broader family of invariant forms in representation theory, culminating in the Casimir Operator, which generalizes the Killing form to arbitrary representations. The form's invariance under all automorphisms makes it a natural tool for classifying real forms of complex Lie algebras and for studying the geometry of symmetric spaces. Its definition depends fundamentally on the Adjoint Representation of the Lie algebra on itself, making the Killing form a self-referential construction: the algebra measures its own structure through its action on itself.