Crystallography, relativity and octonions

In condensed matter theory many invaluable models rely on the possibility of subsuming fundamental particle interactions in constitutive relations for macroscopic fields in near equilibrium assemblies of particles. Should one wish to maintain relativistic covariance this substitution generates a problem that can only be addressed by expanding the dimension of the space-time base manifold (four) to that of its tangent bundle (eight). The linear vector space of the octonions over the real (or conceivably rational) field seem to offer definite advantages in doing this.