# Non left $k$-Engel elements in a nilpoent group always generate this group

Given a finite nilpotent group $G$ and let us denote by $L_n(G)$ the set of left $n$-Engel elements in $G$.

Assume that $n$ is the smallest positive integer such that $L_n(G)=G$.

Is it true that $G$ is always generated by $G-L_{n-1}(G)$?

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