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1. CJM Online first

Elduque, Alberto
 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, idempotentCategories:17A75, 14L15, 17B25, 20G15
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