  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 of type
G_2, whereas it is of type A_2 for trialitarian automorphisms of type II.
