The Eight Cayley–Dickson Doubling Products

Abstract

Abstract. The purpose of this paper is to identify all eight of the basic Cayley-Dickson doubling products. A Cayley-Dickson algebra AN+1 of dimension 2 N+1 consists of all ordered pairs of elements of a CayleyDickson algebra AN of dimension 2 N where the product (a, b)(c, d) of elements of AN+1 is defined in terms of a pair of second degree binomials (f (a, b, c, d), g(a, b, c, d)) satisfying certain properties. The polynomial pair(f, g) is called a 'doubling product.' While A0 may denote any ring, here it is taken to be the set R of real numbers. The binomials f and g should be devised such that A1 = C the complex numbers, A2 = H the quaternions, and A3 = O the octonions. Historically, various researchers have used different yet equivalent doubling products.

An Alternate Cayley-Dickson Product

“…There are eight (and only eight) distinct Cayley-Dickson doubling products [4] which accomplish this. For each of the eight, the conjugate of an ordered pair (a, b) is defined recursively by (a, b) * = (a * , −b)…”

“…Only two of these eight, P 3 and P ⊤ 3 have been investigated. The eight algebras resulting from these products are isomorphic [4] and all have the same elements and the same unit basis vectors e 0 , e 1 , e 2 , · · · , e n , · · · . The basis vectors will be defined below.…”

Periodicity of the Cayley-Dickson twists

“…The unit basis vectors {e k } of Cayley-Dickson algebras may be represented as a twisted group with e 0 as the group identity. For each of the eight Cayley-Dickson doubling products [4] there is a twisting map ω(p, q) : N 2 0 → {±1} (where N 0 represents the non-negative integers) with the property that e p e q = ω(p, q)e p⊕q where ⊕ is a group operation on N 0 consisting of the 'bit-wise exclusive or' of the binary representations of non-negative integers.…”

“…Using the general tree to calculate e p e q A version of the general quaternion tree is developed in [4]. The general tree depicted in Figures 3 and 4 suffices for Cayley-Dickson algebras of any dimension for products P 0 through P 3 .…”

“…Theorem 1. The twists of all eight products satisfy the following: [4] ω(p, 0) = ω(0, p) = 1 for all p. ω(p, p) = −1 for all p > 0.…”