  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.
