Quaternions for 3D beam deflections

I require help to develop and use effectively, quaternions for modeling of beam elements but also in extracting local element forces in a nonlinear finite element analysis. My axis system is different than many examples and I need a work-through to ensure everything is OK.

To extract local element forces from beam elements, rigid body movement needs to be extracted from solved global displacements and angles. Several corotational formulations are available online using new axis definitions and manipulations. I desire this to be done with quaternions. All results will need to pass appropriate unit tests.