Structurable Algebras and Groups of Type E6 and E7

It is well known that every group of type F4 is the automorphism group of an exceptional Jordan algebra, and that up to isogeny all groups of type 1E6 with trivial Tits algebras arise as the isometry groups of norm forms of such Jordan algebras. We describe a similar relationship between groups of type E6 and groups of type E7 and use it to give explicit descriptions of the homogeneous projective varieties associated to groups of type E7 with trivial Tits algebras.