  The Standard Model extension with additional Lorentz violating terms allows
for redefining the equation of motion of a propagating left-handed fermionic
particle. The obtained Dirac-type equation can be embedded in a generalized
Lorentz-invariance preserving-algebra through the definition of Lorentz
algebra-like generators with a light-like preferred axis. The resulting
modification to the fermionic equation of motion introduces some novel
ingredients to the phenomenological analysis of the cross section of the
tritium $\beta$-decay. Assuming lepton number conservation, our formalism
provides a natural explanation for the tritium $\beta$-decay end-point via an
effective neutrino mass term without the need of a sterile right-handed state.
