Remark 4
Tag 01G6, permanent. Cite it as [Bourbaki, Tag 01G6].
Lie Groups and Lie Algebras, chapter IX, § 7. Irreducible representations of connected compact Lie groups
The definitions and statements of Chap. VIII, §7, nos. 2 to 5 can be generalized without difficulty to the present situation; we leave the details to the reader.
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.
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Nothing.