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That's not true. Rotations in 3D space are 4D objects: a 3D orientation and a 1D magnitude.

Rotors use a.b with aXb, and quaternions use magnitude and 3D direction vector.

SO(3) is definitely three-dimensional, and rotations don't involve magnitudes.

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

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