Definition 4

Tag 09ZW, permanent. Cite it as [Bourbaki, Tag 09ZW].

Lie Groups and Lie Algebras, chapter I, § 6. Semi-simple Lie algebras

A Lie algebra is called reductive if its adjoint representation is semi-simple.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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