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The second invariant of the left Cauchy-Green deformation tensor $\mathbf{B}$ (or right $\mathbf{C}$) has been argued to play a fundamental role in nonlinear elasticity. Generalized neo-Hookean materials, which depend only on the first invariant, lead to universal relations that conflict with experimental data, fail to display important mechanical behaviors (such as the Poynting effect in simple shear), and may not provide a satisfactory link with the mesoscale. However, the second invariant term is not a higher order strain contribution to the energy, which lead us to reflect on what is incomplete about neo-Hookean materials. Instead of the usual Cauchy-Green elastic formulation, we investigate this matter from the perspective of left stretch $\mathbf{V}= \sqrt{\mathbf{B}}$ and Bell strain $\mathbf{E}_{\text{Bell}} = \mathbf{V}-\mathbf{I}$ formulations. Invariants of these tensors offer a different interpretation than those of $\mathbf{B}$ and are linked to different classes of materials. The main example we adopt is a general isotropic energy quadratic in Bell strains, the quadratic-Biot material. Despite being quadratic in stretch like neo-Hookean, this material presents both the classic and reverse Poynting effect in simple shear, whose direction switches as a function of the constant conjugate to the second invariant of $\mathbf{E}_{\text{Bell}}$. Its second normal stress also presents a local maximum as a function of the amount of shear, a transition that is not observed in a Mooney-Rivlin solid. Moreover, even the Varga model, linear in Bell strains, presents Poynting in simple shear, which poses the question of why this is not true for a model linear in Green-Lagrange strains. Pure torsion of a solid cylinder is also discussed, particularly how the behavior of the resultant axial force contrasts between the different formulations.