Quaternion
Commutative subrings

The relationship of quaternions to each other within the complex subplanes of H {\displaystyle \mathbb {H} } can also be identified and expressed in terms of commutative subrings. Specifically, since two quaternions p and q commute (i.e., p q = q p) only if they lie in the same complex subplane of H {\displaystyle \mathbb {H} }, the profile of H {\displaystyle \mathbb {H} } as a union of complex planes arises when one seeks to find all commutative subrings of the quaternion ring.
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