Right 4-Engel elements of a group

We prove that the set of right 4-Engel elements of a group $G$ is a subgroup for locally nilpotent groups $G$ without elements of orders 2, 3 or 5; and in this case the normal closure $<x>^G$ is nilpotent of class at most 7 for each right 4-Engel elements $x$ of $G$.
Comment: to appear in Journal of Algebra and its Applications; This paper has some overlaps with arXiv:0906.2439, however the proofs are clearer and arXiv:0906.2439 will not publish anywhere