  For those finite-matrix representations of the Lorentz group of
rotations/boosts with spin $(A,B)\oplus(C,D)$ that can also represent
translations, two possible translation subgroups qualify. Of these two, one
must be selected, and one discarded, to represent the Poincar\'{e} group of
rotations/boosts with translations in spacetime. Instead, let us discard the
requirement that there be just one translation subgroup. With
dual-translations, one gives up agreement with simple macroscopic observations
of spacetime. Now the transformations of both possible translation subgroups
combine with those of the Lorentz group. The resulting commutation relations
require new transformations and generators to satisfy the linearity requirement
of a Lie algebra. Special cases of spins are sought to restrict the influx of
new transformations. One finds that the Dirac 4-spinor formalism is the only
viable solution. The slightly expanded group it represents is the conformal
group with just one new transformation, scale change. It follows as a corollary
that the Dirac 4-spinor formalism is the only matrix representation of the
conformal group with spin $(A,B)\oplus(C,D).$
