  Let $g$ be an element of a finite group $G$ and let $R_{n}(g)$ be the
subgroup generated by all the right Engel values $[g,{}_{n}x]$ over $x\in G$.
In the case when $G$ is soluble we prove that if, for some $n$, the Fitting
height of $R_{n}(g)$ is equal to $k$, then $g$ belongs to the $(k+1)$th Fitting
subgroup $F_{k+1}(G)$. For nonsoluble $G$, it is proved that if, for some $n$,
the generalized Fitting height of $R_n(g)$ is equal to $k$, then $g$ belongs to
the generalized Fitting subgroup $F^*_{f(k,m)}(G)$ with $f(k,m)$ depending only
on $k$ and $m$, where $|g|$ is the product of $m$ primes counting
multiplicities. It is also proved that if, for some $n$, the nonsoluble length
of $R_n(g)$ is equal to $k$, then $g$ belongs to a normal subgroup whose
nonsoluble length is bounded in terms of $k$ and $m$. Earlier similar
generalizations of Baer's theorem (which states that an Engel element of a
finite group belongs to the Fitting subgroup) were obtained by the first two
authors in terms of left Engel-type subgroups.
