Choosing the Rotation Angle

What angle to choose? From the point of view of parsimony we can reduce the number of variables from 13 to 12 by rotating our coordinate system around the 3-axis by an angle = $(\arctan (2c_{45}/(c_{55}-c_{44})))/2$. This will anihilate the two occurences of the c₄₅ term.  

$$\pmatrix{c_{11}&c_{12}&c_{13}&c_{16}&0&0\cr c_{12}&c_{22}&c... ...c_{36}&c_{66}&0&0\cr 0&0&0&0&c_{44}&0\cr 0&0&0&0&0&c_{55}\cr}$$ (4)

Does this make sense physically? Yes. Since there is 2-fold symmetry, the three waves propagating along the 3-axis have orthogonal directions of particle motion, and those directions corresponding to the two pure shear waves define two pseudo-symmetry axes. If the coordinate system is now aligned along these axes, the c45 term disappears,