Automorphism invariance and identities
  If an outer (multilinear) commutator identity holds in a large subgroup of a group, then it holds also in a large characteristic subgroup. Similar assertions are valid for algebras and their ideals or subspaces. Varying the meaning of the word "large", we obtain many interesting facts. These results cannot be extended to arbitrary (non-multilinear) identities. As an application, we give a sharp estimate for the `virtual derived length' of (virtually solvable)-by-(virtually solvable) groups. 
Evgenii I. Khukhro, Anton A. Klyachko, Natalia Yu. Makarenko, Yulia B.   Melnikova