Lie Groups Beyond an Introduction
《Lie Groups Beyond an Introduction》 · 第 374 页 · Chapter VI，Section 4「Iwasawa Decomposition」，Theorem 6.46
Theorem 5.18.1. Let $A, B$ be two subalgebras of the algebra $\operatorname{End} E$ of endomorphisms of a finite dimensional vector space $E$, such that $A$ is semisimple and $B = \operatorname{End}_A E$. Then: (i) $A = \operatorname{End}_B E$ (i.e., the centralizer of the centralizer of $A$ is $A$); (ii) $B$ is semisimple; (iii) As a representation of $A\otimes B$, $E$ decomposes as $E = \bigoplus_{i\in I} V_i \otimes W_i$, where $V_i$ are all the irreducible representations of $A$ and $W_i$ are all the irreducible representations of $B$.