The algebra of turns

It follows that a turn is the geometric realization of a normalized quaternion, that the addition of turns is the geometric realization of the multiplication of quaternions of unit norm, and that the group of turns is isomorphous with the group of normalized quaternions. Furthermore, eqs. (1) and (2) provide the correct parameterization of a normalized quaternion as 

We have now introduced four parameterizations for SO(3) R(o n) (Section 2.1) R(a, 3, 7), where a, f3, 7 are the three Euler angles (Section 11.7) R(a, b), where a, b are the complex Cayley-Klein parameters (Section 11.6) and the Euler-Rodrigues parameterization [X A], 

Exercise 12.5-4 Show that two rotations through infinitesimally small angles commute. 

One immediate application of the quaternion formulae (19) and (20) for the multiplication of rotations is the proof that all rotations through the same angle are in the same class. To find the rotations in the same class as R(d n) we need to evaluate 

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