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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