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We produce an explicit formula for the dispersion relation for the DiracEquation in the Standard Model Extension (SME) in the presence of Lorentzviolation. Our expression is obtained using a novel techniques which exploitthe algebra of quaternions. The dispersion relation is found to convenientlyfactor in two special cases that each involve a mutually exclusive set ofnon-vanishing Lorentz-violating parameters. This suggests that a usefulapproach to studies of Lorentz-violating models is to split the parameter spaceinto two separate pieces, each of which yields a simple, tractable dispersionrelation that can be used for analysis.

Factoring the Dispersion Relation in the Presence of Lorentz Violation