complexification

《Class Field Theory》 · 印刷第 65 页 Section 3.6「The Complexification of a Real Lie Algebra」，Definition 3.36 与 Proposition 3.37 完整陈述（省略两者之间的记号说明）

Definition 3.36. If $V$ is a finite-dimensional real vector space, then the complexification of $V$, denoted $V_{\mathbb C}$, is the space of formal linear combinations $v_1+iv_2$, with $v_1,v_2\in V$. This becomes a real vector space in the obvious way and becomes a complex vector space if we define $i(v_1+iv_2)=-v_2+iv_1$. […] Proposition 3.37. Let $\mathfrak g$ be a finite-dimensional real Lie algebra and $\mathfrak g_{\mathbb C}$ its complexification. Then the bracket operation on $\mathfrak g$ has a unique extension to $\mathfrak g_{\mathbb C}$ that makes $\mathfrak g_{\mathbb C}$ into a complex Lie algebra. The complex Lie algebra $\mathfrak g_{\mathbb C}$ is called the complexification of the real Lie algebra $\mathfrak g$.