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Section 2.5 Octonions

What happens if we add another, independent, square root of \(-1\text{?}\) We get a new number system called the octonions.

(The material in this section was originally published in [1].)

The day after his discovery of the quaternions in October 1843, Hamilton sent a letter to his good friend John T. Graves. On 26 December 1843, Graves wrote back describing the octonions, which he called octaves, a name that is still sometimes used. However, Graves didn't publish this work until 1845, shortly after (and in response to) Arthur Cayley's publication of his own discovery of the octonions. For this reason, the octonions are also known as Cayley numbers. Although Hamilton later vouched for Graves' priority, Cayley did publish first; both are given credit for independently discovering the octonions.

In analogy to the previous construction of \(\CC\) and \(\HH\text{,}\) an octonion \(x\) can be thought of as a pair of quaternions, \((x_{\HH1},x_{\HH2})\text{,}\) so that

\begin{equation} \OO = \HH \oplus \HH \, \ell\tag{2.5.1} \end{equation}

Since we are running out of letters, we will denote \(i\) times \(\ell\) simply as \(i\ell\text{,}\) and similarly with \(j\) and \(k\text{.}\) But what about the remaining products?

Of course, \(\ell^2=-1\text{;}\) this is built into the construction. It is easy to see that \(i\ell\text{,}\) \(j\ell\text{,}\) and \(k\ell\) also square to \(-1\text{;}\) there are now seven independent imaginary units, and we could write

\begin{equation} x = x_1 + x_2 i + x_3 j + x_4 k + x_5 k\ell + x_6 j\ell + x_7 i\ell + x_8 \ell\tag{2.5.2} \end{equation}

where \(x_m\in\RR\text{,}\) which can be thought of as a point or vector in \(\RR^8\text{.}\) The real part of \(x\) is just \(x_1\text{;}\) the imaginary part of \(x\) is everything else. Algebraically, we could define

\begin{align} \Re(x) \amp= \frac12 (x+\bar{x}) ,\tag{2.5.3}\\ \Im(x) \amp= \frac12 (x-\bar{x})\tag{2.5.4} \end{align}

where it is important to note that the imaginary part is, well, imaginary. This differs slightly from the standard usage of these terms for complex numbers, where “\(\Im(z)\)” normally refers to a real number, the coefficient of \(i\text{.}\) This convention is not possible here, since the imaginary part has seven degrees of freedom, and can be thought of as a vector in \(\RR^7\text{.}\)

Figure 2.5.1. The octonionic multiplication table.

The full multiplication table is summarized in Figure 2.5.1. Each point corresponds to an imaginary unit. Each line corresponds to a quaternionic triple, much like \(\{i,j,k\}\text{,}\) with the arrow giving the orientation. For example,

\begin{align} k~\ell \amp= k\ell ,\tag{2.5.5}\\ \ell~k\ell \amp= k ,\tag{2.5.6}\\ k\ell~k \amp= \ell\tag{2.5.7} \end{align}

and each of these products anticommutes, that is, reversing the order contributes a minus sign.

A remarkable property of the octonions is that they are not associative! For example, compare

\begin{align} (i~j) (\ell) \amp= +(k)(\ell) ~=~ +k\ell ,\tag{2.5.8}\\ (i) (j~\ell) \amp= (i)(j\ell) ~=~ -k\ell .\tag{2.5.9} \end{align}

However, the octonions are alternative, that is, products involving no more than 2 independent octonions do associate. Specifically,

\begin{align} (xy)y \amp= xy^2 ,\tag{2.5.10}\\ (xy)x \amp= x(yx)\tag{2.5.11} \end{align}

for any octonions \(x\text{,}\) \(y\text{.}\) Alternativity extends to products with conjugates, so that

\begin{align} (xy)\bar{y} \amp= x|y|^2 ,\tag{2.5.12}\\ (xy)\bar{x} \amp= x(y\bar{x}) .\tag{2.5.13} \end{align}