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For proofs of this and the following theorems see my paper in Transactions, loc. cit. Transactions of the American Mathematical Society, Vol. 3.
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Hawkes, H.E. On quaternion number-systems. Math. Ann. 60, 437–447 (1905). https://doi.org/10.1007/BF01457622
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DOI: https://doi.org/10.1007/BF01457622