  Let $R_n(G)$ denotes the set of all right $n$-Engel elements of a group $G$.
We show that in any group $G$ whose 5th term of lower central series has no
element of order 2, $R_3(G)$ is a subgroup. Furthermore we prove that $R_4(G)$
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 $x\in R_4(G)$. Using a group constructed by Newman and Nickel
we also show that, for each $n\geq 5$, there exists a nilpotent group of class
$n+2$ containing a right $n$-Engel element $x$ and an element $a\in G$ such
that both $[x^{-1},_n a]$ and $[x^{k},_n a]$ are of infinite order for all
integers $k\geq 2$. We finish the paper by proving that at least one of the
following happens: (1) There is an infinite finitely generated $k$-Engel group
of exponent $n$ for some positive integer $k$ and some 2-power number $n$. (2)
There is a group generated by finitely many bounded left Engel elements which
is not an Engel group.
