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Section 2.7 The Cayley–Dickson Process

(Some of the material in 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{2.7.1} \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{2.7.2} \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{2.7.3} \end{equation}

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

\begin{equation} q = (z,w)\tag{2.7.4} \end{equation}

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

\begin{equation} \bar{(a,b)} = (\bar{a},-b)\tag{2.7.5} \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{2.7.6} \end{equation}

so that

\begin{equation} (a,b)\bar{(a,b)} = (|a|^2+|b|^2,0)\tag{2.7.7} \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{2.7.8} \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{2.7.9} \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.

Turning this notation around, the Cayley–Dickson process can be summarized as follows.

  • Begin with an algebra \(\KK\) possessing an operation of conjugation (\(a\longmapsto\bar{a}\)) and norm (\(a\longmapsto|a|^2=a\bar{a}\)).

  • Extend \(\KK\) to the vector space \(\KK\oplus\KK e\text{.}\)

  • Extend conjugation on \(\KK\) to \(\KK\oplus\KK e\) by requiring

    \begin{equation} \bar{ae}=-ae\tag{2.7.10} \end{equation}
    for all \(a\in\KK\text{.}\)

  • Extend the product on \(\KK\) to \(\KK\oplus\KK e\) by requiring

    \begin{align} a(de) \amp= (da)e\notag\\ (be)c \amp= (b\bar{c})e\tag{2.7.11}\\ (be)(de) \amp= -\epsilon\,\bar{d}b\notag \end{align}
    from which it follows that \(e^2=-\epsilon=\mp1\) and \(ec=\bar{c}e\text{.}\)