11.88 (1990)

We define the length $l(g)$ of an Engel element $g$ of a group $G$ to be the smallest number $l$ such that $[h, g; l] = 1$ for all $h \in G$. Here $[h, g; 1] = [h, g]$ and $[h, g; i + 1] = [[h, g; i], g]$. Does there exist a polynomial function $\phi(x, y)$ such that $l(uv) \leqslant \phi(l(u), l(v))$? Up to now, it is unknown whether a product of Engel elements is again an Engel element.