Corollary 3
Tag 09X2, permanent. Cite it as [Bourbaki, Tag 09X2].
Lie Groups and Lie Algebras, chapter I, § 4. Nilpotent Lie algebras
Let V be a vector space and g a finite-dimensional subalgebra of whose elements are nilpotent endomorphisms of V. Then g is a nilpotent Lie algebra.
This follows immediately from Lemma 1 and Corollary 1.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.