  Using complexified quaternions, an intriguing link between generators of two
different and surprisingly commuting four-dimensional representations of the
SU(2)xU(1) Lie group, and generators of two four-dimensional spin 1/2
representations of the Spin(3,1) Lie group is established: the former
generators completely determine the latter ones, and cross-combined they
constitute two different, but closely related, four-dimensional representations
of Spin(3,1)xSU(2)xU(1). These representations are used to construct a
Spin(3,1)xSU(2)xU(1) gauge invariant Lagrangian, containing two four-spinors
consisting not as usual of Weyl two-spinors of opposite helicity and equal weak
isospin, but instead of Weyl two-spinors of opposite weak isospin and equal
helicity, a construction which arises naturally from the mathematical formalism
itself. A possible future generalization, using complexified octonions, is
discussed.
