A note on conciseness of Engel words

It is still an open problem to determine whether the nth Engel word [x,_ n y] is concise, that is, if for every group G such that the set of values e_n (G) taken by [x,_n y] on G is finite it follows that the verbal subgroup E_n (G) generated by e n (G) is also finite. We prove that if e_n (G) is finite, then [E_n (G), G] is finite, and either G/[E_n (G), G] is locally nilpotent and E_n (G) is finite, or G has a finitely generated section that is an infinite simple n-Engel group. It follows that [x,_n y] is concise if n is at most four.