Exercise 11, § 5

With the notations of no. 2, assume that $k$ is non-discrete and locally compact, hence isomorphic to $\mathbf{R},\mathbf{C}$ or a finite extension of $\mathbf{Q}_p($Commutative Algebra, Chap. VI, §9, no. 3). For all $n\geq 1$, the quotient $k^*/k^{*n}$ is finite (cf. Commutative Algebra, Chap. VI, §9, Exerc. 3 for the ultrametric case). Deduce that the quotients $T_Q/$Im(T$_P$) and Aut($\mathfrak{g}$)$/$Aut$_e(\mathfrak{g})$ are finite.

When $k=\mathbf{R}$, show that $T_Q/$Im(T$_P$) is isomorphic to the dual of the $\mathbf{F}_2$-vector space $(Q\cap 2P)/2Q$. When $k=\mathbf{C}$, we have $T_Q=$ Im(T$_P$); this is the integral subgroup of the Lie group Aut($\mathfrak{g}$) with Lie algebra $\mathfrak{h}$.