### The Cayley-Dickson Process

We have constructed the complex numbers, the quaternions, and the octonions by doubling a smaller algebra. We have \begin{align} \CC &= \RR \oplus \RR i \\ \HH &= \CC \oplus \CC j \\ \OO &= \HH \oplus \HH \ell \end{align} We can emphasize this doubling, using a slightly different notation. A complex number $z$ is equivalent to pair of real numbers, its real and imaginary parts. So we can write \begin{equation} z = (x,y) \end{equation} corresponding in more traditional language to $z=x+iy$. Conjugation and complex multiplication then become \begin{align} \bar{(a,b)} &= (a,-b) \\ (a,b)(c,d) &= (ac-bd,bc+ad)\\ (a,b)\bar{(a,b)} &= (a^2+b^2,0) \end{align} A quaternion $q$ can be written as a pair of complex numbers, \begin{equation} q = (z,w) \end{equation} corresponding to $q=z+wj$. Conjugation now takes the form \begin{equation} \bar{(a,b)} = (\bar{a},-b) \end{equation} but what about quaternionic multiplication? Working out $(a+bj)(c+dj)$ with $a,b,c,d\in\CC$, we see that \begin{equation} (a,b)(c,d) = (ac-b\bar{d},ad+b\bar{c}) \end{equation} so that \begin{equation} (a,b)\bar{(a,b)} = (|a|^2+|b|^2,0) \end{equation} Finally, if we write an octonion $p$ as two quaternions, corresponding to $p=q+r\ell$, we obtain \begin{align} \bar{(a,b)} &= (\bar{a},-b) \\ (a,b)(c,d) &= (ac-\bar{d}b,da+b\bar{c}) \\ (a,b)\bar{(a,b)} &= (|a|^2+|b|^2,0) \end{align}

All of the above constructions are special cases of the Cayley-Dickson process, for which \begin{align} \bar{(a,b)} &= (\bar{a},-b) \\ (a,b)(c,d) &= (ac-\epsilon\bar{d}b,da+b\bar{c}) \\ (a,b)\bar{(a,b)} &= (|a|^2+\epsilon|b|^2,0) \end{align} where $\epsilon=\pm1$. We can use this construction to generate larger algebras from smaller ones, by making successive choices of $\epsilon$ at each step.

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