Definition 6

Tag 0A0E, permanent. Cite it as [Bourbaki, Tag 0A0E].

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

A special automorphism of $g$ is an automorphism of $g$ of the form $e^{\mathrm{ad}\, x}$ , where $x$ is in the nilpotent radical of $g$ .

Clearly a special automorphism leaves every ideal of $g$ stable.

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

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Nothing.