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.

20G15, 11E57, 17A75, 14L10

algebraic group of type D_4, triality, outer automorphism of order 3, composition algebra, symmetric composition, octonions, okubo algebra