Proposition 9
Tag 05DJ, permanent. Cite it as [Bourbaki, Tag 05DJ].
Algebra, chapter I, § 6. Extensions, solvable groups, nilpotent groups
Let G be a nilpotent group of class ≤ n and let N be a normal subgroup of G. There exists a series of subgroups
such that (G, N^k) ⊂ N^{k+1} for k = 1, ..., n.
If (G^k) satisfies condition (b) of Proposition 7, then take
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.