Groups of type $\mathrm{E}_6$ and $\mathrm{E}_7$ over Rings via Brown Algebras and Related Torsors

Abstract: We study structurable algebras and their associated Freudenthal triple systems over commutative rings. The automorphism groups of these triple systems are exceptional groups of type $\mathrm{E}_7$, and we realize groups of type $\mathrm{E}_6$ as centralizers. When 6 is invertible, we further give a geometric description of homogeneous spaces of type $\mathrm{E}_7/\mathrm{E}_6$, and show that they parametrize principal isotopes of Brown algebras. As opposed to the situation over fields, we show that such isotopes may be non-isomorphic.