  The purpose of this paper is to identify all of the Cayley-Dickson doubling
products. A Cayley-Dickson algebra $\mathbb{A}_{N+1}$ of dimension $2^{N+1}$
consists of all ordered pairs of elements of a Cayley-Dickson algebra
$\mathbb{A}_{N}$ of dimension $2^N$ where the product $(a,b)(c,d)$ of elements
of $\mathbb{A}_{N+1}$ is defined in terms of a pair of second degree binomials
$\left(f(a,b,c,d),g(a,b,c,d)\right)$ satisfying certain properties. The
polynomial pair$(f,g)$ is called a `doubling product.' While $\mathbb{A}_{0}$
may denote any ring, here it is taken to be the set $\mathbb{R}$ of real
numbers. The binomials $f$ and $g$ should be devised such that
$\mathbb{A}_{1}=\mathbb{C}$ the complex numbers, $\mathbb{A}_{2}=\mathbb{H}$
the quaternions, and $\mathbb{A}_{3}=\mathbb{O}$ the octonions . Historically,
various researchers have used some but not all of these doubling products.
