  Let $G$ be a group. We denote by $\nu(G)$ a certain extension of the
non-abelian tensor square $G \otimes G$ by $G \times G$. In this paper we prove
that the derived subgroup $\nu(G)'$ is a central product of three normal
subgroups of $\nu(G)$, all isomorphic to the non-abelian tensor square $G
\otimes G$. As a consequence, we describe the structure of each term of the
derived and lower central series of the group $\nu(G)$.
