On the right n-Engel group elements

In this paper we study right $n$-Engel group elements. By modifying a group constructed by Newman and Nickel, we construct, for each integer $ngeq 5$, a 2-generator group $G =langle a, brangle$ with the property that $b$ is a right $n$-Engel element but where $[b^k,_n a]$ is of infinite order when $knotin {0, 1}$.