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Section 1.5 The Cayley–Dickson Process

(This section was originally published in [1].)

We have constructed the complex numbers, the quaternions, and the octonions by doubling a smaller algebra. We have

\begin{equation} \begin{aligned} \CC \amp= \RR \oplus \RR i \\ \HH \amp= \CC \oplus \CC j \\ \OO \amp= \HH \oplus \HH \ell\end{aligned}\tag{1.5.1} \end{equation}

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)\tag{1.5.2} \end{equation}

corresponding in more traditional language to \(z=x+iy\text{.}\) Conjugation and complex multiplication then become

\begin{equation} \begin{aligned} \bar{(a,b)} \amp= (a,-b) \\ (a,b)(c,d) \amp= (ac-bd,bc+ad)\\ (a,b)\bar{(a,b)} \amp= (a^2+b^2,0)\end{aligned}\tag{1.5.3} \end{equation}

A quaternion \(q\) can be written as a pair of complex numbers,

\begin{equation} q = (z,w)\tag{1.5.4} \end{equation}

corresponding to \(q=z+wj\text{.}\) Conjugation now takes the form

\begin{equation} \bar{(a,b)} = (\bar{a},-b)\tag{1.5.5} \end{equation}

but what about quaternionic multiplication? Working out \((a+bj)(c+dj)\) with \(a,b,c,d\in\CC\text{,}\) we see that

\begin{equation} (a,b)(c,d) = (ac-b\bar{d},ad+b\bar{c})\tag{1.5.6} \end{equation}

so that

\begin{equation} (a,b)\bar{(a,b)} = (|a|^2+|b|^2,0)\tag{1.5.7} \end{equation}

Finally, if we write an octonion \(p\) as two quaternions, corresponding to \(p=q+r\ell\text{,}\) we obtain

\begin{equation} \begin{aligned} \bar{(a,b)} \amp= (\bar{a},-b) \\ (a,b)(c,d) \amp= (ac-\bar{d}b,da+b\bar{c}) \\ (a,b)\bar{(a,b)} \amp= (|a|^2+|b|^2,0)\end{aligned}\tag{1.5.8} \end{equation}

All of the above constructions are special cases of the Cayley-Dickson process, for which

\begin{equation} \begin{aligned} \bar{(a,b)} \amp= (\bar{a},-b) \\ (a,b)(c,d) \amp= (ac-\epsilon\bar{d}b,da+b\bar{c}) \\ (a,b)\bar{(a,b)} \amp= (|a|^2+\epsilon|b|^2,0)\end{aligned}\tag{1.5.9} \end{equation}

where \(\epsilon=\pm1\text{.}\) 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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