  Let $w=w(x_1,\ldots,x_r)$ be an outer commutator word. We show that the word
$w(u_1,\ldots,u_r)$ is concise whenever $u_1,\ldots,u_r$ are non-commutator
words in disjoint sets of variables. This applies in particular to words of the
form $w(x_1^{n_1},\ldots,x_r^{n_r})$, where the $n_i$ are non-zero integers.
Our approach is via the study of values of $w$ on normal subgroups, and in this
setting we obtain the following result: if $N_1,\ldots,N_r$ are normal
subgroups of a group $G$ and the set of all values $w(g_1,\ldots,g_r)$ with
$g_i\in N_i$ is finite then also the subgroup generated by these values, i.e.
$w(N_1,\ldots,N_r)$, is finite.
