Algebraic groups of type D4, triality, and composition algebras

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