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Let $w=w(x_1,\ldots,x_r)$ be a lower central word or a derived word. We showthat the word $w(u_1,\ldots,u_r)$ is concise whenever $u_1,\ldots,u_r$ arenon-commutator words in disjoint sets of variables, thus proving a generalizedversion of a conjecture of Azevedo and Shumyatsky. This applies in particularto words of the form $w(x_1^{n_1},\ldots,x_r^{n_r})$, where the $n_i$ arenon-zero integers. Our approach is via the study of values of $w$ on normalsubgroups, 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 subgroupgenerated by these values, i.e. $w(N_1,\ldots,N_r)$, is finite.

Conciseness on normal subgroups and new concise words from lower central and derived words