If we consider two vectors in $\Re ^3$, $\vec{v_1} = (1,0,1)$ and $\vec{v_2} = (0,1,1)$ (represented as (3,0,3) and (0,3,3) for clarity) and operate on them with a linear transformation that preserves their length but rotates them $\frac{\pi}{2}$ radians, we can see that our $\vec{v_1}' = \vec{v_2}$ and our $\vec{v_2}' = \vec{v_1}$ with a negative x component.

Based on the analysis in the paper in question, we are now able to assign a "rotation vector" as our transformation of $\vec{v_1}$.

(1)It therefore follows that our transformation of $\vec{v_2}$ can be shown as the transformation of $\vec{v_1}$ transformed by a rotation of 90 degrees. We must, however, include a column in matrix $T_{\frac{\pi}{2}}$ because our Z component is preserved regardless of the rotation.

(2)Compiling all of these vectors into an A matrix that we can call our rotational transformation,

(3)which looks very similar to the two dimensional rotation matrix, but this one preserves the sign and value of the z component which makes sense if the vectors are rotating about the z axis.