Lie Groups Beyond an Introduction
《Lie Groups Beyond an Introduction》 · 第 36 页 · Chapter I, Section 3「Field Extensions and the Killing Form」，式 (1.18)
Now let $\mathfrak g$ be a finite-dimensional Lie algebra over $k$. If $X$ and $Y$ are in $\mathfrak g$, then $\operatorname{ad}X\operatorname{ad}Y$ is a linear transformation from $\mathfrak g$ to itself, and it is meaningful to define $B(X,Y)=\operatorname{Tr}(\operatorname{ad}X\operatorname{ad}Y)$. Then $B$ is a symmetric bilinear form on $\mathfrak g$ known as the Killing form of $\mathfrak g$ after the person who introduced it.