Abstract.A condition for when two eight-dimensional absolute valued algebras are isomorphic was given in [4]. We use this condition to deduce a description (in the sense of Dieterich, [9]) of the category of such algebras, and show how previous descriptions of some full subcategories fit in this description. Led by the structure of these examples, we aim at systematically constructing new subcategories whose classification is manageable. To this end we propose, in greater generality, the definition of sharp stabilizers for group actions, and use these to obtain conditions for when certain subcategories of groupoids are full. This we apply to the category of eight-dimensional absolute valued algebras and obtain a class of subcategories, for which we simplify, and partially solve, the classification problem.

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We study exceptional Jordan algebras and related exceptional group schemes over commutative rings from a geometric point of view, using appropriate torsors to parametrize and explain classical and new constructions, and proving that over rings, they give rise to non-isomorphic structures.We begin by showing that isotopes of Albert algebras are obtained as twists by a certain F 4 -torsor with total space a group of type E 6 , and using this, that Albert algebras over rings in general admit non-isomorphic isotopes, even in the split case as opposed to the situation over fields. We then consider certain D 4 -torsors constructed from reduced Albert algebras, and show how these give rise to a class of generalised reduced Albert algebras constructed from compositions of quadratic forms. Showing that this torsor is non-trivial, we conclude that the Albert algebra does not uniquely determine the underlying composition, even in the split case. In a similar vein, we show that a given reduced Albert algebra can admit two coordinate algebras which are nonisomorphic and have non-isometric quadratic forms, contrary, in a strong sense, to the case over fields, established by Albert and Jacobson.

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Any group of type F4 is obtained as the automorphism group of an Albert algebra. We prove that such a group is R-trivial whenever the Albert algebra is obtained from the first Tits construction. Our proof uses cohomological techniques and the corresponding result on the structure group of such Albert algebras.