Right Engel-type subgroups and length parameters of finite groups

Let be an element of a finite group and let be the subgroup generated by all the right Engel values over. In the case when is soluble we prove that if, for some, the Fitting height of is equal to, then belongs to the th Fitting subgroup. For nonsoluble, it is proved that if, for some, the generalized Fitting height of is equal to, then belongs to the generalized Fitting subgroup with depending only on and, where is the product of primes counting multiplicities. It is also proved that if, for some, the nonsoluble length of is equal to, then belongs to a normal subgroup whose nonsoluble length is bounded in terms of and. 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.