A nilpotency criterion for some verbal subgroups

The word $w=[x_{i_1},x_{i_2},\dots ,x_{i_k}]$ is a simple commutator word if $k\ge 2,i_1\mathrm{\ne }{i}_2$ and $i_j\in \{1,\dots ,m\}$, for some $m>1$. For a finite group $G$, we prove that if $i_1\mathrm{\ne }{i}_j$ for every $j\mathrm{\ne }1$, then the verbal subgroup corresponding to $w$ is nilpotent if and only if $\mathrm{\mid }ab\mathrm{\mid }=\mathrm{\mid }a\mathrm{\mid }\mathrm{\mid }b\mathrm{\mid }$ for any $w$-values $a,b\in G$ of coprime orders. We also extend the result to a residually finite group $G$, provided that the set of all $w$-values in $G$ is finite.