Lemma 7

Tag 03F0, permanent. Cite it as [Bourbaki, Tag 03F0].

Théories spectrales, chapter V, § 4. Représentations des groupes compacts

Let $H_2$ be a real Lie group and let $H_1$ be a closed subgroup of $H_2$ . The complexified Lie algebra of $H_1$ identifies with a subrepresentation of the adjoint representation of $H_1$ on the complexified Lie algebra of $H_2$ .

In fact, the restriction to $H_1$ of the adjoint representation of $H_2$ on its Lie algebra identifies with the adjoint representation of $H_1$ .

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Nothing.