  The classification of all fourth order anisotropic tensor classes for
classical linear elasticity is well known. We review the mathematical procedure
behind this classification and extend it to fourth order elasticity tensors
acting on non-symmetric matrices. These tensors naturally appear in generalised
continuum models. Based on tensor-symmetrization we provide the most general
form of these tensors for the case of orthotropic, transversely isotropic,
cubic and isotropic materials. We give a self-contained presentation and
provide a detailed calculation for simple examples.
