Missouri Journal of Mathematical Sciences

An Alternate Cayley-Dickson Product

John W. Bales

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Although the Cayley-Dickson algebras are twisted group algebras, little attention has been paid to the nature of the Cayley-Dickson twist. One reason is that the twist appears to be highly chaotic and there are other interesting things about the algebras to focus attention upon. However, if one uses a doubling product for the algebras different from yet equivalent to the ones commonly used, and if one uses a numbering of the basis vectors different from the standard basis a quite beautiful and highly periodic twist emerges. This leads easily to a simple closed form equation for the product of any two basis vectors of a Cayley-Dickson algebra.

Article information

Missouri J. Math. Sci., Volume 28, Issue 1 (2016), 88-96.

First available in Project Euclid: 19 September 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 16S99: None of the above, but in this section
Secondary: 16W99: None of the above, but in this section

Cayley-Dickson doubling product twisted group product fractal twist tree


Bales, John W. An Alternate Cayley-Dickson Product. Missouri J. Math. Sci. 28 (2016), no. 1, 88--96. doi:10.35834/mjms/1474295358. https://projecteuclid.org/euclid.mjms/1474295358

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  • W. Ambrose, Structure theorems for a special class of Banach algebras, Trans. Amer. Math. Soc., 57 (1945), 364–386.
  • J. Baez, The Octonions, Bul. Am. Math. Soc., 39.2 (2001), 145–205.
  • J. Bales, A tree for computing the Cayley-Dickson twist, Missouri J. of Math. Sci., 21.2 (2009), 83–93.
  • J. Bales, The eight Cayley-Dickson doubling products, Adv. in Appl. Clifford Alg., 25.4 (2015), 1–23.
  • D. Biss, J. Christensen, D. Dugger, and D. Isaksen, Eigentheory of Cayley-Dickson algebras in Forum Mathematicum, 21.5 (2009), 833–851.
  • R. Brown, On generalized Cayley-Dickson algebras, Pacific J. Math., 20.3 (1967), 415–422.
  • R. Busby and H. Smith, Representations of twisted group algebras, Trans. Am. Math. Soc., 149.2 (1970), 503–537.
  • C. Flaut and V. Shpakivskyi, Holomorphic Functions in Generalized Cayley-Dickson Algebras, Springer Basel, Adv. in Appl. Clifford Algebras, 25.1 (2015), 95–112.
  • L. E. Dickson, On quaternions and their generalization and the history of the eight square theorem, Annals of Mathematics 20.3 (1919), 155–171.
  • C. M. Edwards and J. T. Lewis, Twisted group algebras I, Communications in Mathematical Physics, 13.2 (1969), 119–130.
  • W. F. Reynolds, Twisted group algebras over arbitrary fields, Illinois J. Math., 15.1 (1971), 91–103.
  • R. Schafer, On the algebras formed by the Cayley-Dickson process, Amer. J. Math., 76 (1954), 435–446.


  • This article has been retracted by the author.: A Retraction. Missouri J. Math. Sci. 32 (2020), no. 1, 118. doi:10.35834/2020/3201118.