Killing form

Summary

The Killing form of a finite-dimensional Lie algebra $\mathfrak g$ is the symmetric bilinear form $\kappa(x, y) = \tr(\operatorname{ad} x \circ \operatorname{ad} y)$. It is an invariant bilinear form that every finite-dimensional Lie algebra carries. By Cartan's criterion, in characteristic $0$ it detects solvability and semisimplicity: $\mathfrak g$ is semisimple exactly when $\kappa$ is nondegenerate. For the Lie algebra of a compact Lie group it is negative semidefinite.