Lemma 3

Tag 0AFM, permanent. Cite it as [Bourbaki, Tag 0AFM].

Lie Groups and Lie Algebras, chapter III, § 6. Real and complex Lie groups

Let $G$ be a Lie group, $H_1$ and $H_2$ integral subgroups whose topology admits a countable base and $g \in G$ . Then

g H_1 g^{-1} = H_2 \Leftrightarrow (\mathrm{Ad}\, g)(L(H_1)) = L(H_2).

$\mathrm{Ad}\, g = T_e(\mathrm{Int}\, g)$ . Hence, by transport of structure, $(\mathrm{Int}\, g)(H_1)$ has Lie algebra $(\mathrm{Ad}\, g)(L(H_1))$ . As $H_1$ and $H_2$ have countable bases, to say that the sets $H_2$ and $(\mathrm{Int}\, g)(H_1)$ are equal amounts to saying that the integral subgroups $H_2$ and $(\mathrm{Int}\, g)(H_1)$ are equal (no. 2, Corollary 3 to Proposition 3). Then the lemma follows from Theorem 2 (i).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Proposition 14, § 6

Cites

Proposition 3
Theorem 2