Obtaining the Gauge Invariant Kinetic Term for a SU(n)${}_U$ x SU(m)${}_V$ Lagrangian

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.