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 $\mathbb{C}h\_\eta\oplus\frak{g}\_{-\eta}\oplus\frak{g}\_\eta$.



A property of compact involutions of semi-simple Lie algebras?
Actually the proof is just one line: because
$$
[\frak{h}\oplus\frak{g}\_\alpha\oplus\frak{g}\_{-\alpha}, \frak{h}\oplus\frak{g}\_\alpha\oplus\frak{g}\_{-\alpha}]\subset \mathbb{C}h\_\alpha\oplus\frak{g}\_\alpha\oplus\frak{g}\_{-\alpha}.
$$


As noted by Allen, this works for any root and any map preserving the Lie bracket.


