In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) show that Killing form has a close relationship to the semisimplicity of the Lie algebras.

The Killing form was essentially introduced into Lie algebra theory by Élie Cartan (1894) in his thesis. In a historical survey of Lie theory, Borel (2001) has described how the term "Killing form" first occurred in 1951 during one of his own reports for the Séminaire Bourbaki; it arose as a misnomer, since the form had previously been used by Lie theorists, without a name attached. Some other authors now employ the term "Cartan-Killing form".^{[citation needed]} At the end of the 19th century, Killing had noted that the coefficients of the characteristic equation of a regular semisimple element of a Lie algebra are invariant under the adjoint group, from which it follows that the Killing form (i.e. the degree 2 coefficient) is invariant, but he did not make much use of the fact. A basic result that Cartan made use of was Cartan's criterion, which states that the Killing form is non-degenerate if and only if the Lie algebra is a direct sum of simple Lie algebras.

Consider a Lie algebra ${\displaystyle {\mathfrak {g}}}$ over a field K. Every element x of ${\displaystyle {\mathfrak {g}}}$ defines the adjoint endomorphism ad(x) (also written as ad_x) of ${\displaystyle {\mathfrak {g}}}$ with the help of the Lie bracket, as

Now, supposing ${\displaystyle {\mathfrak {g}}}$ is of finite dimension, the trace of the composition of two such endomorphisms defines a symmetric bilinear form

with values in K, the Killing form on ${\displaystyle {\mathfrak {g}}}$.

The following properties follow as theorems from the above definition.

Given a basis e_i of the Lie algebra ${\displaystyle {\mathfrak {g}}}$, the matrix elements of the Killing form are given by

Here