 Full frame indifference degenerates a quadratic small-strain energy

Write a quadratic small-strain energy as $\tilde{W}(F)=Q(\mathrm{sym}(F-I))$ with $Q$ homogeneous of degree two. If it were fully frame indifferent, the rotation $R=\mathrm{diag}(1,-1,-1)$ would have zero energy. Quadratic homogeneity would then make every $F_s=I+s(R-I)$ have zero energy. At $s=1/4$, $F_s=\mathrm{diag}(1,1/2,1/2)$ has determinant $1/4$, yet zero energy. Thus a nondegenerate linear elasticity energy cannot also be used as a fully objective finite-deformation energy. Its invariance concerns infinitesimal rotations around the reference state.