Corollary 2
Tag 058R, permanent. Cite it as [Bourbaki, Tag 058R].
Algebra, chapter I, § 4. Groups and groups with operators
Let G be a group and X a subset of G consisting of pairwise permutable elements. The subgroup of G generated by X is commutative.
The set consists of pairwise permutable elements (§ 2, no. 3, Proposition 5) and the law induced on the stable subset generated by Y is commutative (§ 1, no. 5, Corollary 2).
If G is a group with operators, the stable subgroup generated by a subset of G consisting of pairwise permutable elements is not necessarily commutative.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.