The general form of the one- and two-point correlation tensor of a homogeneous and (K, θ) -isotropic random field and the spectral expansion of such a field in terms of stochastic integrals with respect to certain random measures depend on the choice of a basis in the linear space where the field takes its values. We choose a basis for 11 different fields. It turns out that the basis depends only on the crystal system of the group K.