  Order three elements in the exceptional groups of type G2 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 G2 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.
