It is classical to take a division algebra over $\mathbb{R}$ and defining an H-space structure on the unit spheres by restricting and normalizing.

There are commutative division algebras of dimension 1 and 2 leading to commutative products on $S^0$ and $S^1$ identifying them as Eilenberg-MacLane spaces - Or if we forget some structure as an $E_{\infty}$-spaces.

The associative division algebras $\mathbb{H}$ defines an associative product on $S^3$, which is also a Lie-group, but forgetting some structure it is an $A_\infty$-space.

There division algebra $\mathbb{O}$ defines an $A_2$ structure on $S^7$, which is not $A_\infty$ (is it $A_3$?).

As is well known it is possible to prove that no other spheres has $A_2$ structure.

**Question:** Is there a heiraki of structures below $A_2$ yet related such that $S^{15}$ has this structure, but $S^{31}$ does not?

Remark: A heiraki below $A_2$ could be that $A_2=D_\infty$ for some definition of structures $D_n$, analagous to $E_1$ being $A_\infty$.

**Question:** Is there an even more general definition of "lower" structures and a statement about all spheres (including possibly non-trivial structures on even-dimensional spheres)?