  Let $G$ be a finite soluble group and $G^{(k)}$ the $k$th term of the derived
series of $G$. We prove that $G^{(k)}$ is nilpotent if and only if
$|ab|=|a||b|$ for any $\delta_k$-values $a,b\in G$ of coprime orders. In the
course of the proof we establish the following result of independent interest:
Let $P$ be a Sylow $p$-subgroup of $G$. Then $P\cap G^{(k)}$ is generated by
$\delta_k$-values contained in $P$. This is related to the so-called Focal
Subgroup Theorem.
