Structurable algebras and groups of type E6 and E7

It is well-known that every group of type F 4 is the automorphism group of an exceptional Jordan algebra, and that up to isogeny all groups of type 1 E 6 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 E 6 and groups of type E 7 and use it to give explicit descriptions of the homogeneous projective varieties associated to groups of type E 7 with trivial Tits algebras. We also show that the kernel of the Rost invariant for quasi-split groups of type E 6 and E 7 is trivial. It is well-known that over an arbitrary field F (which for our purposes we will assume has characteristic = 2, 3) every algebraic group of type F 4 is obtained as the automorphism group of some 27-dimensional exceptional Jordan algebra and that some groups of type E 6 can be obtained as automorphism groups of norm forms of such algebras. In [Bro63], R.B. Brown introduced a new kind of F-algebra, which we will call a Brown algebra. The automorphism groups of Brown algebras provide a somewhat wider class of groups of type E 6 , specifically all of those with trivial Tits algebras. Allison and Faulkner [AF84] showed that there is a Freudenthal triple system (i.e., a quartic form and a skew-symmetric bilinear form satisfying certain relations) determined up to similarity by every Brown algebra. The automorphism group of this triple system is a simply connected group of type E 7 , and we show that this provides a construction of all simply connected groups of type E 7 with trivial Tits algebras (and more generally all Freudenthal triple systems). This is interesting because it allows one to relate properties of these algebraic groups over our ground field F , which are generally hard to examine, with properties of these algebras, which are relatively much easier to study. Brown algebras are neither power-associative nor commutative, but they do belong to a wide class of algebras with involution known as central simple structurable algebras which were introduced in [All78], see [All94] for a nice survey. Other examples of such algebras are central simple associative algebras with involution and Jordan algebras. Brown algebras comprise the most poorly understood class of central simple structurable algebras. Although they are simple and so by definition have …