Algebraic groups of type $D_4$, triality, and composition algebras.

Abstract

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.