Corollary 2

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

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

Let G be a Lie group and H1 and H_2 integral subgroups of G. If $L(H_2) \subset L(H_1)$ , then H_2 is an integral subgroup of H_1.*

Let $i_1 : H_1 \to G, i_2 : H_2 \to G$ be the canonical injections. Then

L(i_2)(L(H_2)) = L(H_2) \subset L(H_1).

By Proposition 1,

i_2

is an analytic mapping of

H_2

into

H_1

and even an immersion of

H_2

into

H_1

since

L(i_2)

is an isomorphism of

L(H_2)

onto a subalgebra of

L(H_1)

admitting a topological supplement.

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

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