Proposition 9

Tag 01AQ, permanent. Cite it as [Bourbaki, Tag 01AQ].

Lie Groups and Lie Algebras, chapter IX, § 2. Maximal tori of compact Lie groups

Let G be a connected compact Lie group, T a maximal torus of G, H a Lie group and $f: G\rightarrow H$ a morphism. Then $f$ is injective if and only if its restriction to T is injective.

Indeed, by Th. 2 (no. 2) the normal subgroup Ker $f$ of G reduces to the identity element if and only if its intersection with T reduces to the identity element.

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

Proposition 6, § 4

Cites

Th. 2