Conjugacy classes of trialitarian automorphisms and symmetric compositions

The trialitarian automorphisms considered in this paper are the outer automorphisms of order 3 of adjoint classical groups of type D-4 over arbitrary fields. A one-to-one correspondence is established between their conjugacy classes and similarity classes of symmetric compositions on 8-dimensional quadratic spaces. Using the known classification of symmetric compositions, we distinguish two conjugacy classes of trialitarian automorphisms over algebraically closed fields. For type I, the group of fixed points is connected and simple of type G(2), whereas, if the characteristic of the field is different from 3, it is adjoint of type A(2) for trialitarian automorphisms of type II.