### Inner Automorphisms of the Octonions

#### Posted by John Baez

What are the inner automorphisms of the octonions?

Of course this is an odd question. Since the octonions are nonassociative you might fear the map

$f : \mathbb{O} \to \mathbb{O}$

given by

$f(x) = g x g^{-1}$

for some octonion $g \ne 0$ is *not even well-defined!*

But it is.

The reason is that the octonions are **alternative**: the unital subalgebra generated by any two octonions is associative. Furthermore, the inverse $g^{-1}$ of $g \ne 0$ is in the unital subalgebra generated by $g$. This follows from

$g^{-1} = \frac{1}{|g|^2} \overline{g}$

and the fact that $\overline{g}$ is in the unital subalgebra generated by $g$, since we can write $g = a + b$ where $a$ is a real multiple of the identity and $b$ is purely imaginary, and then $\overline{g} = a - b = 2a - g$.

It follows that whenever $g$ is a nonzero octonion, we have

$(g x) g^{-1} = g (x g^{-1})$

for all octonions $x$, so we can write either as

$f(x) = g x g^{-1} .$

However, there is no reason *a priori* to expect $f$ to be an automorphism, meaning

$f(xy) = f(x) f(y)$

for all $x,y \in \mathbb{O}$. For which octonions $g$ does this happen?

Of course it happens when $g$ is real, i.e. a real multiple of $1$. But that’s boring—because then $f$ is the identity. Can we find more *interesting* inner automorphisms of the octonions?

A correspondent, Charles Wynn, told me that $f$ is an automorphism when

$g = \frac{1}{2} + \frac{\sqrt{3}}{2} i$

and $i \in \mathbb{O}$ is any element with $i^2 = -1$. This kind of element $g$ is a particular sort of 6th root of unity in the octonions—one that lies at a $60^\circ$ angle from the positive real axis.

A bit of digging revealed this paper:

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

In Theorem 2.1, Lamont claims that $f(x) = g x g^{-1}$ is an automorphism of the octonions iff and only if $g$ is either real or

$(\mathrm{Re}(g))^2 = \frac{1}{4} |g|^2.$

In other words, $f$ is an automorphism iff the octonion $g$ lies at an angle of $0^\circ, 60^\circ, 120^\circ$ or $180^\circ$ from the positive real axis. These cases include *all* 6th roots of unity in the octonions!

I haven’t fully checked the proof, but it seems to use little more than the Moufang identity.

I wonder what this fact means? How do these inner automorphisms sit inside the group of *all* automorphisms of the octonions, $\mathrm{G}_2$?

## Re: Inner Automorphisms of the Octonions

How many 6th roots of unity in the octonions are there?

Are there any general results about numbers of roots of octonion polynomials?