I asked this question on MSE but didn't receive any answer so far. My question is the following:

Let $G$ be a compact connected Lie group and $ \sigma :G \rightarrow G $ be an involution on $G$. Let $G^\sigma :=\lbrace g \in G, \sigma(g)=g \rbrace$.

Denote by $G_\mathbb{C}$ the complexification of $G$. Does there exist an involution $\tilde{\sigma}$ on $G_\mathbb{C}$ which coïncides with $\sigma$ on $G$ and such that ${(G^{\sigma})}_\mathbb{C} = {(G_\mathbb{C})}^\tilde{\sigma}$ ?

My though was to define $\tilde{\sigma}$ to be $\sigma$ on G and to be the identity on the complement of $G$ on $G_\mathbb{C}$, but don't see whether this imply the ${(G^{\sigma})}_\mathbb{C} = {(G_\mathbb{C})}^\tilde{\sigma}$ or not, any help please!