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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.

[[Category:Mathematics]]