What are the inner automorphisms of the octonions?

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

π: π β π

given by

π(π₯) = ππ₯πβ»ΒΉ

for a nonzero octonion π isn't well-defined! After all, maybe we have

(ππ₯)πβ»ΒΉ β π(π₯πβ»ΒΉ)

But in fact this is a red herring.

(1/n)

(Below we see my friend the physicist Nichol Furey, a big fan of the octonions, standing in front of the Fano plane, which can be used to multiply them.)

Luckily, the octonions are 'alternative': the unital subalgebra generated by any two octonions is associative. Furthermore, the inverse πβ»ΒΉ of any nonzero octonion is in the unital subalgebra generated by π. Thus π, π₯ and πβ»ΒΉ all lie in a unital subalgebra generated by two octonions, so

(ππ₯)πβ»ΒΉ = π(π₯πβ»ΒΉ)

and we can write either one as ππ₯πβ»ΒΉ without fear.

So the big question is whether

π(π₯) = ππ₯πβ»ΒΉ

is an automorphism - that is, whether it obeys π(π₯π¦)=π(π₯)π(π¦).

(2/n)

In other words: if we have a nonzero octonion π, do we have

π(π₯π¦)πβ»ΒΉ = (ππ₯πβ»ΒΉ) (ππ¦πβ»ΒΉ)

for all octonions π₯ and π¦? Again this is not obvious, because the octonions are nonassociative!

And indeed it's not always true.

This paper:

P. J. C. Lamont, Arithmetics in Cayley's algebra, Glasgow Mathematical Journal, 6 no. 2 (1963), 99-106. https://www.cambridge.org/core/services/aop-cambridge-core/content/view/F911E6DABC9FCABB17A531156AD272AF/S2040618500034808a.pdf/div-class-title-arithmetics-in-cayley-s-algebra-div.pdf

claims to settle when it's true! And the answer given is interesting.

(3/n)

Namely, a nonzero octonion π has

π(π₯π¦)πβ»ΒΉ = (ππ₯πβ»ΒΉ) (ππ¦πβ»ΒΉ)

for all octonions π₯ and π¦ precisely when π lies at a 0 degree, 60 degree, 120 degree or 180 degree angle from the positive real line. (We think of the real multiples of 1 β π as a copy of the real line in the octonions.)

The proof is a terse calculation - but I haven't checked it carefully yet.

(4/n)

In particular, π is a 6th root of 1 in the octonions if and only if |π| = 1 and π lies at a 0, 60, 120 or 180 degree angle from the positive real line.

So, if Lamont is correct, 6th roots of 1 give inner automorphisms of the octonions! In fact they give all the inner automorphisms, since rescaling π goesn't change ππ₯πβ»ΒΉ.

1 and -1 are 6th roots of 1 that give trivial inner automorphisms - just the identity. But the rest are nontrivial!

(5/n)

What are the nontrivial inner automorphisms of the octonions like? How do they sit in the group \(\mathrm{G}_2\) consisting of all automorphisms of the octonions? What sort of subgroup do they generate? (I doubt the composite of inner automorphisms is inner in this nonassociative world.)

Lots of questions!

I thank Charles Wynn for pointing me toward these mysteries.

(6/n, n = 6)

@undefined @johncarlosbaez they generate the whole G_2. Lie subgroups of G_2 are well understood. as the set of inner automorphisms you describe is 6-dimensional the only possibilities we need to exclude is that it generates a copy of SU(3) which is 8 dimensional or a copy of SO(4) (all other proper subgroups of G_2 have dimensions <6). It should be easy to see that it's not SO(4) which is also 6-dimensional.

SU(3) can be excluded as follows.

All SU(3) in G_2 are known to come about as follows. G_2 acts on S^6=unit imaginary octonians and various SU(3) in G_2 are isotropy subgroups of different points in S^6. if all inner automorphisms were in an isotropy group of some point v in S^6 that v would be in the center of octonions. This is impossible and hence Inner automorphisms generate all of G_2.

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@johncarlosbaez sorry, I don't know the answer to this although I am sure this is known. you may want to ask on mathoverflow. in any even as I said it's easy to rule out that inner automorphisms of octonions are contained in SO(4). they are both closed 6 manifolds, the former being S^6 and and there is no embedding of S^6 into SO(4) for easy topological reasons (it would have to be a homeomorphism).

@herid - nice! Over on the n-Category Cafe @allenk points out that the conjugacy class of elements of order 2 in Gβ is isomorphic to Gβ/SO(4), and notes that this is also the space of quaternion subalgebras of π.

There should be a direct connection between these two facts. Maybe any quaternion subalgebra of π is fixed by a unique automorphism of order 2, or something like that. (I'm not sure that's quite right.)

https://golem.ph.utexas.edu/category/2022/11/inner_automorphisms_of_the_oct.html#c061775

John Carlos Baez@johncarlosbaez@mathstodon.xyz@herid - okay, that does the job. How do you know SO(4) is a subgroup of Gβ? I feel I should learn how to classify the maximal closed subgroups of a compact simple Lie group, but I don't know how:

https://mathoverflow.net/questions/60315/modern-reference-for-maximal-connected-subgroups-of-compact-lie-groups