Linking electroweak and gravitational generators

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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.