  The weak currents with respect to the diagonal neutral bosons
$Z$,$Z^{\prime}$, and $Z^{\prime\prime}$ of a specific
\textbf{$SU(4)_{L}\otimes U(1)_{Y}$} gauge model are computed in detail for all
the fermion families involved therein. Our algebraical approach, which is based
on the general method of solving gauge models with high symmetries proposed
several years ago by Cot\u{a}escu, recovers in a non-trivial way all the
Standard Model values for current couplings of the traditional leptons and
quarks, and predicts plausible values for those of the exotic fermions in the
model.
