  Truesdell's empirical inequalities are considered essential in various fields
of nonlinear elasticity. However, they are often used merely as a sufficient
criterion for semi-invertibility of the isotropic stress strain-relation, even
though weaker and much less restricting constitutive requirements like the
strict Baker-Ericksen inequalities are available for this purpose. We elaborate
the relations between such constitutive conditions, including a weakened
version of the empirical inequalities, and their connection to bi-coaxiality
and related matrix properties. In particular, we discuss a number of issues
arising from the seemingly ubiquitous use of the phrase "$X,Y$ have the same
eigenvectors" when referring to commuting symmetric tensors $X,Y$.
