Geodesic
Algebraic groups of type $D_4$, triality, and composition algebras.
Documenta mathematica, Tome 18 (2013), pp. 413-468.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: Conjugacy classes of outer automorphisms of order 3 of simple algebraic groups of classical type $D_4$ are classified over arbitrary fields. There are two main types of conjugacy classes. For one type the fixed algebraic groups are simple of type $G_2$; for the other type they are simple of type $A_2$ when the characteristic is different from 3 and are not smooth when the characteristic is 3. A large part of the paper is dedicated to the exceptional case of characteristic 3. A key ingredient of the classification of conjugacy classes of trialitarian automorphisms is the fact that the fixed groups are automorphism groups of certain composition algebras.
Classification : 20G15, 11E57, 17A75, 14L10
Keywords: algebraic group of type D_4, triality, outer automorphism of order 3, composition algebra, symmetric composition, octonions, okubo algebra 
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     title = {Algebraic groups of type $D_4$, triality, and composition algebras.},
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Chernousov, Vladimir; Elduque, Alberto; Knus, Max-Albert. Algebraic groups of type $D_4$, triality, and composition algebras.. Documenta mathematica, Tome 18 (2013), pp. 413-468. http://geodesic.mathdoc.fr/item/DOCMA_2013__18__a33/