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If one has two vector spaces, one quaternion linear on the left and the other quaternion linear on the right, the juxtaposed product is then quaternion linear on both the left and the right. However, this is not an acceptable tensor product since half of the quaternion action on each of the two vector spaces is lost. For more than two vector spaces even a tensor product of this type cannot be defined. We wish to thank Professor Jaques Tits for discussing this subject with us.
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Biedenharn, L.C., Sepunaru, D., Horwitz, L.P. (1980). Quaternionic quantum mechanics and Adler's chromostatics. In: Wolf, K.B. (eds) Group Theoretical Methods in Physics. Lecture Notes in Physics, vol 135. Springer, Berlin, Heidelberg . https://doi.org/10.1007/3-540-10271-X_307
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DOI: https://doi.org/10.1007/3-540-10271-X_307
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