  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.
