#### DOCUMENTA MATHEMATICA,
Vol. Extra Volume: Alexander S. Merkurjev's Sixtieth Birthday (2015), 387-405

** Max-Albert Knus and Jean-Pierre Tignol **
Triality and algebraic groups of type ^3D_4

We determine which simple algebraic groups of type $^3\D$ 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.

2010 Mathematics Subject Classification: 20G15, 11E72, 17A75.

Keywords and Phrases: Algebraic group of outer type $^3\mathsf{D}_4$, triality, outer automorphism
of order $3$, composition algebra, symmetric composition, cyclic composition,
octonions, Okubo algebra.

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