  Let $G$ be a group and let $x\in G$ be a left $3$-Engel element of order
dividing $60$. Suppose furthermore that $\langle x\rangle^{G}$ has no elements
of order $8$, $9$ and $25$. We show that $x$ is then contained in the locally
nilpotent radical of $G$. In particular all the left $3$-Engel elements of a
group of exponent $60$ are contained in the locally nilpotent radical.
