= Two compatible twin systems for opposite simple-shear variants
{title2=$U_\pm=I\pm\delta e_3\otimes e_1$}
The unrotated variants differ by $2\delta e_3\otimes e_1$, giving <normal> $e_1$. The <rotation> about $e_2$ with $\cos\theta=(1-\delta^2)/(1+\delta^2)$ and $\sin\theta=2\delta/(1+\delta^2)$ gives $U_+-\overline R U_-=[2\delta/(1+\delta^2)](e_1+\delta e_3)\otimes e_3$, giving the second <normal> $e_3$. If these are the only two compatible <rotations>, the only gradients in $SO(3)U_+$ compatible with $\overline R U_-$ are $U_+$ and $\overline R U_+$. Left multiplication by a <rotation> preserves <rank>, so the second classification follows directly from the first.
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