Exercise 38, § 9

Let $G$ be a finite-dimensional real Lie group and $G_0$ its identity component. Consider the following property: (F) $\mathrm{Ad}(G)$ is closed in $\mathrm{Aut}\, L(G)$. Show that $G$ has property (F) in each of the following cases: (i) $G$ is connected and nilpotent; (ii) every derivation of $L(G)$ is inner; (iii) $G_0$ has property (F) and $G/G_0$ is finite; (iv) $G$ is an upper triangular group; (v) $G_0$ is semi-simple.