Lie Groups – Maximal Compact Subgroups Explained

Question

Let $G$ be a compact (connected) semisimple Lie group. Let $G_\mathbb{C}$ be the complexification of $G$.

Is $G$ a maximal compact subgroup of $G_\mathbb{C}$?

Answer 1

$\DeclareMathOperator\Lie{Lie}\newcommand\g{\mathfrak g}\newcommand\C{{\mathbb C}}$$\g = \Lie(G)$ is maximal among subalgebras of $\g_\C = \Lie(G_\C)$ on which the Killing form is negative definite, so $G$ is a maximal connected, compact subgroup of $G_\C$. According to @YCor, maximal compact subgroups of connected groups are connected, so $G$ is also a maximal compact subgroup.