Killing form

Definition
Suppose $$L$$ is a finite-dimensional Lie algebra over a field $$F$$. The Killing form on $$L$$ is a $$F$$-bilinear form $$\kappa$$ on $$L$$ defined as follows:

$$\kappa(x,y) = \operatorname{tr}(\operatorname{ad}(x) \circ \operatorname{ad}(y))$$.

Here, $$\operatorname{ad}(x)$$ is the adjoint action of $$x$$ on $$L$$ (see Lie ring acts as derivations by adjoint action), viewed as a $$F$$-linear map from $$L$$ to itself. The composite is thus also a $$F$$-linear map from $$L$$ to itself. $$\operatorname{tr}$$ computes the trace of this linear map.

Note that the Killing form can be defined and makes sense only for Lie algebras and not for the more general Lie rings.

Basic properties

 * Killing form is symmetric
 * Associativity-like relation between Killing form and Lie bracket: This states that $$\kappa([x,y],z) = \kappa(x,[y,z])$$ for all $$x,y,z$$ in a Lie algebra.

Cartan's criteria
Cartan's criteria rely on Lie's theorem, which in turn depends on the field being algebraically closed and of characteristic zero.


 * Cartan's first criterion: This states that if $$F$$ is an algebraically closed field of characteristic zero (e.g., the field of complex numbers), then a Lie algebra $$L$$ over $$F$$ is solvable if and only if $$\kappa(x,y) = 0$$ for all $$x \in L$$ and $$y \in [L,L]$$.
 * Cartan's second criterion: This states that if $$F$$ is an algebraically closed field of characteristic zero (e.g., the field of complex numbers), then a Lie algebra over $$F$$ is semisimple if and only if the Killing form is nondegenerate on the algebra.

Other facts

 * Killing form on ideal equals restriction of Killing form
 * Killing form on subalgebra not equals restriction of Killing form
 * Abelian ideal is degenerate for Killing form
 * Nilpotent ideal is degenerate for Killing form