Proposition 11

Assume that $k$ is $\mathbf{R},\mathbf{C}$ or a non-discrete complete ultrametric field. Let $\mathfrak{h}$ be a splitting Cartan subalgebra of $\mathfrak{g}$.

(i) Aut($\mathfrak{g},\mathfrak{h}$) is a Lie subgroup of Aut($\mathfrak{g}$) with Lie algebra ad$\mathfrak{h}$.

(ii) $f(T_Q)$ and $(q\circ f)(T_P)$ are open subgroups of Aut($\mathfrak{g},\mathfrak{h}$).

(iii) Aut$_e(\mathfrak{g})$ is an open subgroup of Aut($\mathfrak{g}$).

(iv) If $k=\mathbf{R}$ or $\mathbf{C}$, Aut$_e(\mathfrak{g})$ is the identity component of Aut($\mathfrak{g}$), in other words Int($\mathfrak{g}$).

By Chap. III, §3, no. 8, Cor. 2 of Prop. 29, and no. 10, Prop. 36, Aut($\mathfrak{g},\mathfrak{h}$) is a Lie subgroup of Aut($\mathfrak{g}$) whose Lie algebra is the set of ad $x(x\in \mathfrak{g})$ such that (ad $x$)$\mathfrak{h}\subset \mathfrak{h}$, in other words ad$\mathfrak{h}$.

Let $H\in \mathfrak{h}$. There exists $\varepsilon >0$ with the following properties: for $t\in k$ and $|t|< \varepsilon$, exp($t\gamma (H)$) is defined for all $\gamma \in P(R)$, and the map $\gamma \rightarrow$ exp($t\gamma (H)$) is a homomorphism $\sigma_t$ from P(R) to $k^*$. For $|t|< \varepsilon$, exp($t$ ad H) is defined, induces the identity on $\mathfrak{h}$ and induces on $\mathfrak{g}^{\alpha}$ the homothety with ratio $\sigma_t(\alpha )$; hence exp $t$ ad $H\in (q\circ f)(T_P)$. This proves, in view of (i), that $(q\circ f)(T_P)$ contains a neighbourhood of 1 in Aut($\mathfrak{g},\mathfrak{h}$), and consequently is an open subgroup of Aut($\mathfrak{g},\mathfrak{h}$). A fortiori$,f(T_Q)$ is an open subgroup of Aut($\mathfrak{g},\mathfrak{h}$).

For all $\alpha \in R$, exp ad $\mathfrak{g}^{\alpha}\subset$ Aut$_e(\mathfrak{g})$. In view of (ii), Aut$_e(\mathfrak{g})$ contains a neighbourhood of 1 in Aut($\mathfrak{g}$), which proves (iii).

Assume that $k=\mathbf{R}$ or $\mathbf{C}$. Then Aut$_e(\mathfrak{g})$ is contained in the identity component C of Aut($\mathfrak{g}$) (Chap. VII, §3, no. 1), and is open in Aut($\mathfrak{g}$) by (iii). Thus Aut$_e(\mathfrak{g}) = C$. Finally, C = Int($\mathfrak{g}$) by Chap. III, §9, no. 8, Prop. 30 (i).