Class Field Theory
《Class Field Theory》 · 第 170 页 · Section 7.1「Semisimple and Reductive Lie Algebras」，Definition 7.2
Definition 7.2. If $\mathfrak g$ is a semisimple Lie algebra, a real subalgebra $\mathfrak k$ of $\mathfrak g$ is a compact real form of $\mathfrak g$ if $\mathfrak k$ is isomorphic to the Lie algebra of some compact matrix Lie group and every element $Z$ of $\mathfrak g$ can be expressed uniquely as $Z=X+iY$, with $X,Y\in\mathfrak k$.