  We propose a generalized way to formally obtain the gauge invariance of the
kinetic part of a field Lagrangian over which a gauge transformation ruled by
an $SU(n)_{U} \otimes SU(m)_{V}$ coupling symmetry is applied. As an
illustrative example, we employ such a formal construction for reproducing the
standard model Lagrangian. This generalized formulation is supposed to
contribute for initiating the study of gauge transformation applied to
generalized $SU(n)_{U} \otimes SU(m)_{V}$ symmetries as well as for
complementing an introductory study of the standard model of elementary
particles.
