The n-Category Café

The second fact is perhaps not very well known. It may even be hard to understand what it means. Though the octonions are nonassociative, for any nonzero octonion $g$ the map

$$\begin{array}{rccl} f \colon & \mathbb{O} &\to& \mathbb{O} \\ & x & \mapsto & g x g^{-1} \end{array}$$

is well-defined, since $(g x)g^{-1} = g(x g^{-1})$, which one can show using the fact that the octonions are alternative. More surprisingly, whenever $g^6 = 1$, this map $f$ is an automorphism of the octonions:

$$f(x+y) = f(x) + f(y) , \qquad f(x y) = f(x) f(y) \qquad \forall x,y \in \mathbb{O}$$

and $f$ has order 3:

$$f(f(f(x))) = x \qquad \forall x \in \mathbb{O}$$

To understand this latter fact, we can look at

• P. J. C. Lamont, Arithmetics in Cayley’s algebra, Glasgow Mathematical Journal 6 no. 2 (1963), 99–106.

Theorem 2.1 here implies that an octonion $g$ with $|g| = 1$ defines an inner automorphism $f \colon x \to g x g^{-1}$ if and only if $x$ has order 6.

However, the result is stated differently there. Paraphrasing somewhat, Lamont’s theorem says that any $g \in \mathbb{O}$ that is not a real multiple of $1 \in \mathbb{O}$ defines an inner automorphism $f \colon x \to g x g^{-1}$ if and only if $g$ obeys

$$4 \mathrm{Re}(g)^2 = |\!g|^2$$

This equation is equivalent to $\mathrm{Re}(g) = \pm \frac{1}{2} |\!g|$, which for $g$ on the unit sphere in $\mathbb{O}$ is equivalent to $g$ lying at either a $60^\circ$ angle or a $120^\circ$ angle from the octonion $1$.

The octonions $\pm 1$ clearly define inner automorphisms. Thus, an octonion $g$ on the unit sphere defines an inner automorphism if and only if its angle from $1$ is $0^\circ, 60^\circ, 120^\circ$ or $180^\circ$. This in turn is equivalent to $g^6 = 1$. The inner automorphism it defines then has order 3, since $g$ and $-g$ define the same inner automorphism.

However, if you look at Lamont’s proof, you’ll see the equation $4 \mathrm{Re}(g)^2 = |\!g|^2$ plays no direct role! Instead, he really uses the assumption that $g^3$ is a real multiple of $1$, which is implied by this equation (as easily shown using what we’ve just seen).

From Lamont’s work, one can see the Moufang identities and the characteristic equation for octonions are what force all inner automorphisms of the octonions to have order 3.

It’s well known that you can construct the octonions using triality. So, an argument giving a positive answer to my question might involve a link between triality and the Moufang identities. Conway and Smith seem to link them in On Quaternions and Octonions. But I haven’t figured out how to get from the outer automorphisms of $\text{Spin}(8)$ to the inner automorphisms of $\mathbb{O}$, or vice versa.