Exercise 3, § 8

Let $\mathfrak{a}$ be a semi-simple Lie algebra, and $s\in$ Aut($\mathfrak{a}$). Show the equivalence of:

(i) $s\in$ Aut$_0(\mathfrak{a})$.

(ii) $s$ operates trivially on the centre of $U(\mathfrak{a})$.

(iii) For all $x\in \mathfrak{a}$, there exists $t\in$ Aut$_0(\mathfrak{a})$ such that $tx=sx$.

(Use Prop. 6 to show that (iii) $=\Rightarrow$ (i).)