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||Order $3$ elements in $G_2$ and idempotents in symmetric composition algebras|
Order three elements in the exceptional groups of type $G_2$
are classified up to conjugation over arbitrary fields. Their
centralizers are computed, and the associated classification
of idempotents in symmetric composition algebras is obtained.
Idempotents have played a key role in the study and classification
of these algebras.
Over an algebraically closed field, there are two conjugacy classes
of order three elements in $G_2$ in characteristic not $3$ and
four of them in characteristic $3$. The centralizers in characteristic
$3$ fail to be smooth for one of these classes.
Keywords:symmetric composition algebra, Okubo algebra, automorphism group, centralizer, idempotent
Categories:17A75, 14L15, 17B25, 20G15