All compact, connected, complex Lie groups are Abelian.

Question

Here I found one interesting result about connected, complex Lie groups. In the 1 answer, the author claims that this fact is a consequence of the maximus modulus principle, but I can not understand why this is so.

Answer 1

Let $G$ be a compact, connected, complex Lie group. If $h\in G$, consider the map $\sigma_h\colon G\longrightarrow G$ defined by $\sigma_h(g)=h^{-1}gh$ (that is, $\sigma_h$ is the conjugation by $h$). Since it is an automorphism of the group $G$, it induces an automorphism $\Sigma_h$ of its Lie algebra $\mathfrak g$. The map from $G$ into $\mathfrak g$ which maps $h$ into $\Sigma_h$ is an analytic map. But its domain (which is $G$) is compact, and therefore its range is compact. But then, by the maximum principle, its range consists of a single point. So, since $\operatorname{id}_{\mathfrak g}$ belongs to the range, we have that, for each $h\in G$, $\Sigma_h=\operatorname{id}_{\mathfrak g}$. Therefore, for each $X\in\mathfrak g$ and each $h\in G$,$$\sigma_h\left(e^X\right)=e^{\Sigma_h(X)}=e^X,$$and therefore there is a neighborhood $V$ of $\operatorname{id}_g$ (namely, $V=\exp(\mathfrak g)$) such that, for each $g\in V$, $\sigma_h(g)=g$. But $G$ is connected, and so it is generated by $V$. So, if $g,h\in G$,$$gh=hh^{-1}gh=h\sigma_h(g)=hg.$$