Local lower central series members are divisibility-closed in nilpotent group

Statement
Suppose $$G$$ is a nilpotent group. Suppose $$i$$ and $$k$$ are positive integers. Then, the $$i^{th}$$ member of the $$k$$-local lower central series, i.e., the subgroup $$\gamma_i^{k-loc}(G)$$, is a divisibility-closed subgroup of $$G$$. Explicitly, this means that if $$p$$ is a prime number such that $$G$$ is a $$p$$-divisible group, $$\gamma_i^{k-loc}(G)$$ is also $$p$$-divisible.

In particular, this also implies that $$\gamma_i^{k-loc}(G)$$ is a powering-invariant subgroup, i.e., if $$p$$ is a prime number such that $$G$$ is $$p$$-powered, so is $$\gamma_i^{k-loc}(G)$$.

Facts used

 * 1) uses::Commutator-verbal implies divisibility-closed in nilpotent group