Order 3 elements in G2 and idempotents in symmetric composition algebras

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.Comment: 33 page

Gradings on symmetric composition algebras

The group gradings on the symmetric composition algebras over arbitrary fields are classified. Applications of this result to gradings on the exceptional simple Lie algebras are considered too.Comment: 37 page

Reductive homogeneous spaces and nonassociative algebras

The purpose of these survey notes is to give a presentation of a classical theorem of Nomizu that relates the invariant affine connections on reductive homogeneous spaces and nonassociative algebras.Comment: 28 pages. The first version constituted the notes of a course given at the CIMPA research school "G\'eom\'etrie diff\'erentielle et alg\`ebres non associatives", held in Marrakech, April 13-24, 201