How to show that the $SU(2)$ has no outer automorphism, while the $SU(k)$ with $k>2$ has an outer automorphism $\mathbf{Z}/2$?
I understand some would say to look at the Dynkin diagram. But this is not very transparent and intuitive enough.
Are there other smart ways?
For example, I know that we can regard outer automorphism of $SU(k)$ as complex conjugation: $$ g \in SU(k) \mapsto g^* \in SU(k).$$ So we need to show whether it is possible to find conjugation action of $x$ on $g$ $$ g \mapsto x g x^{-1} $$ such that $$ x g x^{-1} = g^*. $$
If we prove that is possible with $k=2$ then $SU(2)$ has no outer automorphism If we prove that is impossible, while the $SU(k)$ with $k>2$, then it has an outer automorphism $\mathbf{Z}/2$.