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Quaternions that represent 3D rotations have exactly 3 parameters, because they are normalized: https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotati...

Quaternions that are not normalized do not form a representation of the spin group Spin(3).

I'm assuming Spin(3) is equivalent to SO(3)?

It's a double cover of SO(3) (because q and -q correspond to the same element of SO(3)), and is isomorphic to SU(2)

Oh wow thanks I understand