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Submission: 1999, Apr. 25

It is well-known that every algebraic group of type F_4 is the automorphism group of an exceptional Jordan algebra, and that up to isogeny all groups of type ^1E_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. In a final section, we use what we have learned to show that the Rost invariant taking values in H^3 has trivial kernel for quasi-split simply connected groups of type E_6 and E_7.

1991 Mathematics Subject Classification: 17A30 (Primary) 11E72, 14L30, 17A40, 17B25, 17C30, 20G15 (Secondary)

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