Boris Rosenfeld claimed that the 128-dimensional compact Riemannian symmetric space on which $\mathrm{E}_8$ acts as isometries could be seen as the "octooctonionic projective plane", $(\mathbb{O} \otimes \mathbb{O})\mathrm{P}^2$. This is the symmetric space called EVIII by Cartan.

This claim has always been hard to understand, since $\mathbb{O} \otimes \mathbb{O}$ is not a division algebra. Nonetheless there seems to be something to it, as witnessed by the magic square.

John Huerta pointed out to me that if $(\mathbb{O} \otimes \mathbb{O})\mathrm{P}^2$ really is something like an octooctonionic projective plane, we could hope it contains a lot of "octooctonionic projective lines". We aren't sure what an "octooctonionic projective line" should be, but one naive guess is that it's $\mathbb{O} \otimes \mathbb{O}$ together with a point at infinity — and thus, a 64-sphere.

Of course there are lots of 64-spheres smoothly embedded in $(\mathbb{O} \otimes \mathbb{O})\mathrm{P}^2$, but here we want 'nice' ones, raising this question:

• What are the maximal totally geodesic spheres in the 128-dimensional compact Riemannian symmetric space EVIII?

For comparison, note that the octonionic projective plane $\mathbb{O}\mathrm{P}^2$ is the 16-dimensional compact Riemannian symmetric space called FII by Cartan. This contains a lot of totally geodesic 8-spheres, which can rigorously be seen as copies of $\mathbb{O}\mathrm{P}^1$.

I've come across a paper that claims to classify all maximal totally geodesic spheres in compact symmetric spaces:

• Tadashi Nagano and Makiko Sumi, The spheres in symmetric spaces, Hokkaido Math. J. 20(2) (1991), 331–352.

However, it doesn't seem to discuss the case of EVIII.