Introduction to Mathematical Logic, Sixth Edition
《Introduction to Mathematical Logic, Sixth Edition》 · 第 354–355 页 · Chapter 7，Section 7.3.1「Compact Form and Maximal Compact Torus」
A fundamental result of Weyl asserts that one can choose elements $X_\alpha \in \mathfrak g_\alpha$ for each $\alpha \in \Phi$ such that the following holds: • The triple $\{X_\alpha, X_{-\alpha}, H_\alpha\}$ satisfies $[X_\alpha, X_{-\alpha}] = H_\alpha$. • The linear transformation $\theta \colon \mathfrak g \to \mathfrak g$ given by $\theta(X_\alpha) = -X_{-\alpha}$ for $\alpha \in \Phi$ and $\theta(h) = -h$ for $h \in \mathfrak h$ is a Lie algebra automorphism. • The subspace $\mathfrak u = \sum_{\alpha \in \Phi} \mathbb R i H_\alpha + \sum_{\alpha \in \Phi} \mathbb R (X_\alpha - X_{-\alpha}) + \sum_{\alpha \in \Phi} \mathbb R i(X_\alpha + X_{-\alpha})$ is a real Lie algebra, and the restriction of $B$ to $\mathfrak u \times \mathfrak u$ is negative definite.