Lie Groups Beyond an Introduction
《Lie Groups Beyond an Introduction》 · 第 49–50 页 · Chapter I, Section 7「Cartan's Criterion for Semisimplicity」的底域约定及 Proposition 1.46（省略中间其他命题）
In this section, $k$ denotes a subfield of $\mathbb C$, and $\mathfrak g$ denotes a finite-dimensional Lie algebra over $k$. […] Proposition 1.46 (Cartan's Criterion for Solvability). The Lie algebra $\mathfrak g$ is solvable if and only if its Killing form $B$ satisfies $B(X,Y)=0$ for all $X\in\mathfrak g$ and $Y\in[\mathfrak g,\mathfrak g]$.