Corollary 3

Let $a \in \mathbf{L}(G)$. The vector field defined by $a$ and the left operation $g \mapsto \mathrm{Int}\, g$ of $G$ on $G$ is the field $\mathbf{R}_a - \mathbf{L}_a$.

The value of this field at $g$ is

For all $g \in G$ and all $t \in L(G)$,

Since $\mathrm{Ad}\,g = T_e(\mathrm{Int}\,g)$, Proposition 42 of no. 11 proves that $\mathrm{Ad}$ is an analytic linear representation of $G$ on the normable space $L(G)$.