  We determine which simple algebraic groups of type $^3D_4$ over arbitrary
fields of characteristic different from 2 admit outer automorphisms of order 3,
and classify these automorphisms up to conjugation. The criterion is formulated
in terms of a representation of the group by automorphisms of a trialitarian
algebra: outer automorphisms of order 3 exist if and only if the algebra is the
endomorphism algebra of an induced cyclic composition; their conjugacy classes
are in one-to-one correspondence with isomorphism classes of symmetric
compositions from which the induced cyclic composition stems.
