I need to prove the statement below. Since my background on Lie theory is rather weak, I post it here.
Let $\frak{g}$ be a complex semi-simple Lie algebra. Fix a Cartan subalgebra $\frak{h}$ with roots $\Delta\subset\frak{h}^*$ and positive roots $\Delta^+\subset\Delta$, so that we have a decomposition $$ \frak{g}=\frak{h}\oplus\bigoplus_{\alpha\in\Delta}\frak{g}_\alpha. $$
Let $\eta\in\Delta^+$ be the longest root and $h_\eta\in\frak{h}$ be the corresponding coroot. Let $\rho: \frak{g}\rightarrow\frak{g}$ be a compact involution globally fixing $\frak{h}\oplus\frak{g}_{-\eta}\oplus\frak{g}_\eta$.
Problem. Prove that $\rho$ globally fixes $\mathbb{C}h_\eta\oplus\frak{g}_{-\eta}\oplus\frak{g}_\eta$.
For $\frak{sl}_n\mathbb{C}$, taking the usual $\frak{h}$ and $\Delta^+$, since $\rho$ corresponds to a hermitian metric on $\mathbb{C}^n$, I can prove the above statement by straightforward computations. Actually, $\rho$ globally fixes $\frak{h}\oplus\frak{g}_{-\eta}\oplus\frak{g}_\eta$ if and only if the corresponding hermitian metric is of the form $$ \begin{pmatrix} *&&&&&*\\ &*&&&&\\ &&*&&&\\ &&&*&&\\ &&&&*&\\ *&&&&&* \end{pmatrix}, $$ then computations show that $\rho$ globally fixes $\mathbb{C}h_\eta\oplus\frak{g}_{-\eta}\oplus\frak{g}_\eta$.