  In this paper we study left and right 4-Engel elements of a group. In
particular, we prove that $<a, a^b>$ is nilpotent of class at most 4, whenever
$a$ is any element and $b^{\pm 1}$ are right 4-Engel elements or $a^{\pm 1}$
are left 4-Engel elements and $b$ is an arbitrary element of $G$. Furthermore
we prove that for any prime $p$ and any element $a$ of finite $p$-power order
in a group $G$ such that $a^{\pm 1}\in L_4(G)$, $a^4$, if $p=2$, and $a^p$, if
$p$ is an odd prime number, is in the Baer radical of $G$.
