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
$F_{f(k,m)}^{\ast }(G)$
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.