Definition 2
Tag 0AJJ, permanent. Cite it as [Bourbaki, Tag 0AJJ].
Lie Groups and Lie Algebras, chapter III, § 9. Commutators, centralizers and normalizers in a Lie group
A connected real or complex Lie group is called semi-simple if it is finite-dimensional and satisfies the conditions of Proposition 26.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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Nothing.