  We produce an explicit formula for the dispersion relation for the Dirac
Equation in the Standard Model Extension (SME) in the presence of Lorentz
violation. Our expression is obtained using a novel techniques which exploit
the algebra of quaternions. The dispersion relation is found to conveniently
factor in two special cases that each involve a mutually exclusive set of
non-vanishing Lorentz-violating parameters. This suggests that a useful
approach to studies of Lorentz-violating models is to split the parameter space
into two separate pieces, each of which yields a simple, tractable dispersion
relation that can be used for analysis.
