Geodesic
Algebraic groups of type $D_4$, triality, and composition algebras.
Documenta mathematica, Tome 18 (2013), pp. 413-468
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Conjugacy classes of outer automorphisms of order 3 of simple algebraic groups of classical type D4​ are classified over arbitrary fields. There are two main types of conjugacy classes. For one type the fixed algebraic groups are simple of type G2​; for the other type they are simple of type A2​ 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.
DOI : 10.4171/dm/405
Classification : 11E57, 14L10, 17A75, 20G15
Mots-clés : algebraic group of type D_4, triality, outer automorphism of order 3, composition algebra, symmetric composition, octonions, okubo algebra 
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     author = {Vladimir Chernousov and Alberto Elduque and Max-Albert Knus and Jean-Pierre Tignol},
     title = {Algebraic groups of type $D_4$, triality, and composition algebras.},
     journal = {Documenta mathematica},
     pages = {413--468},
     year = {2013},
     volume = {18},
     doi = {10.4171/dm/405},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/405/}
}
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Vladimir Chernousov; Alberto Elduque; Max-Albert Knus; Jean-Pierre Tignol. Algebraic groups of type $D_4$, triality, and composition algebras.. Documenta mathematica, Tome 18 (2013), pp. 413-468. doi: 10.4171/dm/405

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