Definition 2

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

Lie Groups and Lie Algebras, chapter I, § 1. Definition of Lie algebras

Let g be a Lie algebra and x an element of g. The linear mapping $y \mapsto [x, y]$ of g into g is called the adjoint linear mapping of x and is denoted by $\operatorname{ad}_g x$ or $\operatorname{ad} x$ .

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

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