[Boris Rosenfeld](https://en.wikipedia.org/wiki/Freudenthal_magic_square#Rosenfeld_projective_planes) 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](https://en.wikipedia.org/wiki/Symmetric_space#Classification_result) 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](https://en.wikipedia.org/wiki/Freudenthal_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](Cayley_plane) $\mathbb{O}\mathrm{P}^2$ is the 16-dimensional compact Riemannian symmetric space called [FII](https://en.wikipedia.org/wiki/Symmetric_space) 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](https://projecteuclid.org/journals/hokkaido-mathematical-journal/volume-20/issue-2/The-sheres-in-symmetric-spaces/10.14492/hokmj/1381413846.full),  *Hokkaido Math. J.* **20**(2) (1991), 331–352.

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