This work is concerned with implementing the hybridizable discontinuous Galerkin (HDG) method to solve the linear anisotropic elastic equation in the frequency domain, focusing in particular on providing a compact description of the discrete problem and an optimal choice of stabilization in defining the HDG numerical traces. Voigt notation is employed in the description of the discrete problem in order to facilitate matrix operation and to provide efficient book-keeping of physical parameters. Additionally, a first-order formulation working with the compliance elasticity tensor is employed to allow for parameter variation within a mesh cell, for better representation of complex media. We determine an optimal choice of stabilization by constructing a hybridized Godunov-upwind flux for anisotropic elasticity possessing three distinct wave speeds. This stabilization removes the need to choose judiciously scaling factors and can be used as a versatile choice for all materials. Its optimality is established by comparing with identity matrix-based stabilization in a wide range of values for the scaling factor. Numerical investigations are carried out in two and three dimensions, for isotropic elasticity and material with varying degree of anisotropy