I'm reading about the special unitary Lie algebras, and seen it said that complex conjugation is not an inner automorphism of $\mathfrak{su}(n)$ for $n>2$. If there an easy way to see this?
I understand that the automorphism would be inner if there is some $g \in SU(n)$ such that $gXg^{-1} = \bar{X}$ for all $X \in \mathfrak{su}(n)$, is there a simple way to see that no such $g$ exists? Thanks very much for any help.