Killing form

Let $𝔤$ be a finite dimensional Lie algebra over a field $k$, and $X,Y\in 𝔤$. Let ${\mathrm{ad}}_X:𝔤\to 𝔤$ be the adjoint action,${\mathrm{ad}}_{X}Y=[X,Y]$.

Then the Killing form on $𝔤$ is a bilinear map

$$B_{𝔤}:𝔤\times 𝔤\to k$$

given by

$$B_{𝔤}(X,Y)=\mathrm{tr}({\mathrm{ad}}_X\circ {\mathrm{ad}}_Y).$$

The Killing form is invariant (http://planetmath.org/InvariantFormLieAlgebras) under the adjoint action and symmetric (since trace issymmetric).