Corollary 4

Let $G$ be a finite-dimensional connected Lie group. (i) $\mathrm{Int}(L(G)) = \mathrm{Ad}(G)$. (ii) Let $Z$ be the centre of $G$. Then $Z$ is a Lie subgroup of $G$ whose Lie algebra is the centre of $L(G)$. The mapping of $G/Z$ into $\mathrm{Int}\, L(G)$ derived from $g \mapsto \mathrm{Ad}\, g$ when passing to the quotient is a Lie group isomorphism. Assertion (i) follows from Corollary 3 (ii) and the remarks after Definition 2. Let $g \in G$. In order that $\mathrm{Ad}\, g = \mathrm{Id}_{L(G)}$, it is necessary and sufficient that $\mathrm{Int}\, g$ coincide with $\mathrm{Id}_G$ on a neighbourhood of $e$ ($§ 4$, no. 1, Theorem 1 (ii)) and hence on the whole of $G$; in other words, it is necessary and sufficient that $g \in Z$. Then (ii) follows from Corollary 1 to Proposition 1.