I miswrote. Rotation calculations are represented (in quaternions and in geometric algebra calculations) as a 3D subspace of R^4, with the "extra" dimension accounted for by constraining quaternions to the unit hypersphere.

Rotations do involve magnitudes when interpreted as an axis (2 dimensional manifold embedded in R^3) and a magnitide (0 to 2pi)

https://en.wikipedia.org/wiki/Axis%E2%80%93angle_representat...

SO(3) is a coordinate-free abstraction that isn't relevant to the topic of GA vs Quaternions for calculating concrete values in computer graphics