Fixed-generator commutator collection in nilpotent groups ($G\prime =[G,a_1]\cdots [G,a_d]$)

If a nilpotent group is generated by $a_1,\dots ,a_d$, every element of its commutator subgroup is a product $[x_1,a_1]\cdots [x_d,a_d]$. The sets in the displayed formula need not be subgroups. Induct on the nilpotency class $c$. Collection in $G/{\gamma }_c(G)$ gives the desired product with a central error. Since ${\gamma }_c(G)=[{\gamma }_{c-1}(G),G]$, that error is a product $[y_1,a_1]\cdots [y_d,a_d]$ with $y_i\in {\gamma }_{c-1}(G)$. These factors are central, and ${\gamma }_{c-1}(G)$ centralizes $G\prime$, so replacing $x_i$ by $x_i{y}_i$ absorbs the error. For a finitely generated pro-p group, applying this formula in all finite quotients and using compactness makes its algebraically generated commutator subgroup closed.