A unified presentation of some properties of the fourth-rank tensor of anisotropic elasticity is given. The unified presentation involves both established concepts such as the Cauchy relations, the Voigt and Reuss bounds, planes of symmetry and specific directions of longitudinal wave propagation, and a new concept, the specific axis for pure shear wave amplitudes. The unified treatment employs the decomposition of the fourth-rank tensor of anisotropic elasticity into two symmetric secondrank tensors and an irreducible, completely symmetric and traceless, fourth-rank tensor. It is shown that a necessary and sufficient condition for a direction to be a normal to a plane of symmetry is that it be both a specific axis and a specific direction.