Corollary 3

Tag 0AD8, permanent. Cite it as [Bourbaki, Tag 0AD8].

Let G be a Lie group and $\phi$ an exponential mapping of G.

(i) There exists a neighbourhood V of 0 in $L(G)$ such that

\mathrm{Ad}(\phi(x)) = \exp \mathrm{ad}\ x.

for all

x \in V

(ii) If $g \in G$ , there exists a neighbourhood W of 0 in $L(G)$ such that

g \phi(x) g^{-1} = (\mathrm{Ad}\ g.x)

for all

x \in W

(i) follows from Corollary 2 and § 3, no. 12, Proposition 44. (ii) follows from Proposition 8 applied to Int g.

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

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