The Eight Cayley Dickson Doubling Products
The purpose of this paper is to identify all eight of the basic Cayley Dickson doubling products. A Cayley Dickson algebra \mathbb{A}_{N+1}$AN+1 of dimension {2^{N+1}$2N+1 consists of all ordered pairs of elements of a Cayley Dickson algebra \mathbb{A}_{N}$AN of dimension {2^N}$2N where the product (a, b)(c, d)(a,b)(c,d) of elements of \mathbb{A}_{N+1}$AN+1 is defined in terms of a pair of second degree binomials (f(a, b, c, d) g(a, b, c,d)(f(a,b,c,d),g(a,b,c,d) satisfying certain properties. The polynomial pair$(f, g)(f,g) is called a 'doubling product. While \mathbb{A}_{0}$A0 may denote any ring, here it is taken to be the set \mathbb{R}$R of real numbers. The binomials {f}$f and {g}$g should be devised such that \mathbb{A}_{1} \mathbb{C}$A1=C the complex numbers, \mathbb{A}_{2} \mathbb{H}$A2=H the quaternions, and \mathbb{A}_{3} \mathbb{O}$A3=O the octonions. Historically, various researchers have used different yet equivalent doubling products.