Let $G$ be a group and let $p$ be a fixed prime. For each positive integer $n$, the $n$-th term of the Jennings-Lazard-Zassenhaus series of the group $G$ is defined by the rule \begin{eqnarray*} D_{n}(G)=\prod_{jp^{k}\geqslant n}\gamma_{j}(G)^{p^{k}} \end{eqnarray*} where for each positive integer $j$, $\gamma_{j}(G)$ denotes de $j$-th term of the lower central series of the group $G$. Then we obtain the series $G=D_{1}(G)\geq D_{2}(G)\geq\ldots$ of charachteristic subgroups of $G$ with the following properties: $[D_{n},D_{m}]\leq D_{n+m}$ and $D_{n}^{p}\leq D_{pn}$ for each $m,n\geqslant 1$.
In a survey of P. Shumyatsky (Applications of Lie ring methods to group Theory) i found the following statement: If a group $G$ is generated by the elements $a_{1},\ldots,a_{m}$, then for each positive integer $n$, $D_{n}$ is generated by $D_{n+1}$ and the elements of the form $[b_{1},\ldots,b_{j}]^{p^{k}}$ where $jp^{k}\geqslant n$ and $b_{1},\ldots,b_{j}\in \{a_{1},\ldots,a_{m}\}$.
Unfortunately i could not prove this statement, even though Shumyatsky has stated in the text that proof can be obtained using some commutator properties.
Does anyone know where I can find explicit proof of this statement?
Ps.1: Link of Shumyatsky's survey: https://arxiv.org/abs/1706.07963
Ps2: The statement is in the page $8$.
