Lie Groups Beyond an Introduction
《Lie Groups Beyond an Introduction》 · 印刷第 369—370 页 Chapter VI Section 4「Iwasawa Decomposition」，Lie 群及 Cartan 分解约定、极大交换子空间与限制根的定义
Throughout this section, $G$ will denote a semisimple Lie group. Changing notation from earlier sections of this chapter, we write $\mathfrak g$ for the Lie algebra of $G$. […] Let $\theta$ be a Cartan involution of $\mathfrak g$ (Corollary 6.18), let $\mathfrak g=\mathfrak k\oplus\mathfrak p$ be the corresponding Cartan decomposition (6.23), and let $K$ be the analytic subgroup of $G$ with Lie algebra $\mathfrak k$. […] Let $\mathfrak a$ be a maximal abelian subspace of $\mathfrak p$. […] For $\lambda\in\mathfrak a^*$, we write $\mathfrak g_\lambda=\{X\in\mathfrak g\mid(\operatorname{ad}H)X=\lambda(H)X\text{ for all }H\in\mathfrak a\}$. If $\mathfrak g_\lambda\ne0$ and $\lambda\ne0$, we call $\lambda$ a restricted root of $\mathfrak g$ or a root of $(\mathfrak g,\mathfrak a)$.