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.
A proper twist on a group G is a function α : G × G → {−1, 1} with the property that, if p, q ∈ G then α(p, q) α(q, q −1 ) = α(pq, q −1 ) and α(p −1 , p) α(p, q) = α(p −1 , pq). The span V of a set of unit vectors B = {ip | p ∈ G} over a ring K with product xy = p,q α(p, q)xpyqipq is a twisted group algebra. If the twist α is proper, then the conjugate defined by x * = p α(p, p −1 )x * p i p −1 and inner product x, y = p xpy * p , satisfy the adjoint properties xy, z = y, x * z and x, yz = xz * , y for all x, y, z ∈ V . Proper twists on Z N 2 over the reals produce the complex numbers, quaternions, octonions and all higher order Cayley-Dickson and Clifford algebras.
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.
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