Remark
Tag 0AIN, permanent. Cite it as [Bourbaki, Tag 0AIN].
Lie Groups and Lie Algebras, chapter III, § 9. Commutators, centralizers and normalizers in a Lie group
Even if H_1 and H_2 are connected Lie subgroups of G, the Lie subalgebra of L(G) generated by [h_1, h_2] is in general distinct from h.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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Cites
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