Proposition 30

(i) Let $\mathfrak{g}$ be a semi-simple real or complex Lie algebra. Then Int $\mathfrak{g}$ is the identity component of Aut $\mathfrak{g}$.. (ii) Let G be a semi-simple connected real or complex Lie group. The adjoint group of G is the identity component of Aut $L(G)$. Its centre reduces to the identity element. Every derivation of $\mathfrak{g}$ is inner (Chapter I, § 6, Corollary 3 to Proposition 1) and hence $L(\mathrm{Int}\,\mathfrak{g}) = L(\mathrm{Aut}\,\mathfrak{g})$, which proves (i). The first assertion of (ii) follows from (i). The second follows from Proposition 29 (iii) and § 6, no. 4, Corollary 4 (ii) to Proposition 10.