\times
SU(2)_B$ to the diagonal subgroup, which we identify as $SU(2)_L$, can occur through the vacuum expectation value of a field in the bifundamental representation of the underlying product group via an appropriately arranged scalar potential. For the purposes of our discussion here we will need only assume that this breaking takes place at a sufficiently high scale, say just below the string compactification scale. As such ideas for product group breaking have been considered in the past [@Barbieri:1994jq], we will not concern ourselves further with this step.

Any additional bifundamental representations will decompose into triplet and singlet representations under the surviving $SU(2)_L$. Gauge invariance of the underlying $SU(2)_A \times SU(2)_B$ theory then [*requires*]{} that the neutrino-mass generating superpotential coupling involving lepton doublets and our $SU(2)$ triplet now be given by $$%
W_{\nu} = (\lambda_{T})_{ij} L_i T L'_j ,
%
\label{WT}$$ where the field $T$ is the $SU(2)_L$ triplet representation and $L$ and $L'$ are two [*different species*]{} of doublets under the $SU(2)_L$ subgroup. That is, we denote by $L$ fields which have representation $(\mathbf{2},1)$ and by $L'$ fields which have representation $(1, \mathbf{2})$ under the original $SU(2)_A
\times SU(2)_B$ gauge theory. These two sets of lepton doublets, each of 