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Its a fairly standard result that one can represent rotations in 3-space with unit quaternions (or pauli matrices which can be mapped to the unit-quaternions).

Working through a related quantum mechanics problem (where we derive the pauli matrices from experimental results and basic postulates of quantum theory) I come to an (expected) ambiguity in the sign of one of the pauli matrices (e.g. there is a $\pm$ in front of one of the matrices). In physics we "establish a phase convention" and pick one over the other, however I think there may be something deeper here.

I'm trying to mathematically explain this ambiguity. Which got me thinking about our standard 3-space parity convention and how it applies to representing rotations with quaternions (and therefore pauli matrices).

Would choosing an opposite parity convention introduce any changes to the quaternions. Or put in another way, would the group representing rotations in left-handed 3-space be different than the one representing rotations in right-handed 3-space?

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