Definition 4

Let $g$ be a Lie $K$-algebra and $M$ a $g$-module. Suppose that $M$, considered as a $K$-module, admits a finite basis. The bilinear form associated with the $g$-module $M$ (or with the corresponding representation) is the symmetric bilinear form $(x, y) \mapsto \operatorname{Tr}(x_M y_M)$ on $g$. If the representation in question is the adjoint representation, the associated bilinear form is called the Killing form of $g$.