Remark 1

Let G be a finite-dimensional connected real or complex Lie group. If G is not semi-simple, G has a commutative connected Lie subgroup $G'$ which is invariant under every continuous automorphism, such that $G' \neq \{e\}$. For let n be the largest nilpotent ideal of $L(G)$; then $n \neq \{0\}$ and the corresponding analytic subgroup N is a Lie subgroup which is invariant under every continuous automorphism of G (no. 7, Proposition 23); the centre $G'$ of N has the desired properties.