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 $G\_\mathbb{C}$?



Question about maximal compact subgroups of Lie groups
$\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](https://mathoverflow.net/questions/433874/question-about-maximal-compact-subgroups-of-lie-groups#comment1117360_433874), maximal compact subgroups of connected groups are connected, so $G$ is also a maximal compact subgroup.


