TSIA. There are four papers by Bhargava on higher composition laws that are publicly available:
- I: A new view on Gauss composition, and quadratic generalizations
- II: On cubic analogues of Gauss composition
- III: The parametrization of quartic rings
- IV: The parametrization of quintic rings
and based on the paper I, there is one more paper in preparation (at that moment) titled "Higher composition laws V: the parametrization of quaternionic and octonionic rings and modules", which I couldn't find. I also checked Bhargava's original thesis on ProQuest, but it only contains materials for I-IV, but not about quaternionic or octonionic rings.
What I recently found is the paper by Gross-Lucianovic and Lucianovic's thesis, which essentially says that the isomorphism classes of quaternionic rings are parametrized by the "ternary quadratic forms" under the twisted $\mathrm{GL}_{3}$-action. So I guess this might be the "quaternionic" part of the "Higher composition law V". However, I checked the papers referred Gross-Lucianovic and Lucianovic, but failed to find any works on parametrizing octonionic rings, and I wonder if there are any (possibly ongoing) works on this.
