  We address the problem, not of the determination -- which usually needs
numerical methods -- but of an accurate analytical estimation of the distance
of a raw elasticity tensor to cubic symmetry and to orthotropy. We point out
that there are not one but several secondorder tensors that carry the likely
cubic/orthotropic coordinate system of the raw tensor. Since all the
second-order covariants of an (exactly) cubic elasticity tensor are isotropic,
distance estimates based only on such covariants are not always accurate. We
extend to cubic symmetry and to orthotropy the technique recently suggested by
Klime\v{s} for transverse isotropy: solving analytically an auxiliary quadratic
minimization problem whose solution is a second-order tensor that carries the
likely cubic coordinate system. Numerical examples are provided, on which we
evaluate the accuracy of different upper bounds estimates of the distance to
cubic or orthotropic symmetry.
