  A BFC-group is a group in which all conjugacy classes are finite with bounded
size. In 1954 B. H. Neumann discovered that if G is a BFC-group then the
derived group G' is finite. Let w=w(x_1,\dots,x_n) be a multilinear commutator.
We study groups in which the conjugacy classes containing w-values are finite
of bounded order. Let G be a group and let w(G) be the verbal subgroup of G
generated by all w-values. We prove that if x^G has size at most m for every
w-value x, then the derived subgroup of w(G) is finite of order bounded by a
function of m and n. If x^{w(G)} has size at most m for every w-value x, then
[w(w(G)),w(G)] is finite of order bounded by a function of m and n.
