An action of the Klein 4-group on the angular velocity

Expressing the angular velocity via Euler angles is a key step, linking kinematics with rigid body dynamics. Once the components of angular velocity are found in a rotating frame, they are (simultaneously) found in an inertial (non-rotating) frame. And once the components are found for successive intrinsic rotations, they are just as readily found for successive extrinsic rotations. The action of the Klein 4-group on the angular velocity, which we describe in this paper, provides further insight into the kinematic relations of rigid body motion, including the critical motion of Dzhanibekov flipping wingnut.