Corollary 2

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

Lie Groups and Lie Algebras, chapter I, § 4. Nilpotent Lie algebras

The Killing form of a nilpotent Lie algebra is zero. For all $x$ and $y$ in a nilpotent Lie algebra $\operatorname{ad} x \circ \operatorname{ad} y$ is nilpotent and hence of zero trace.

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

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