Definition 1
Tag 0AL7, permanent. Cite it as [Bourbaki, Tag 0AL7].
Lie Groups and Lie Algebras, chapter III, § 10. The automorphism group of a Lie group
Let G be a Lie group. An infinitesimal automorphism of G is any analytic vector field on G satisfying the conditions of Lemma 1.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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Nothing.