Classification of division -graded alternative algebras

The octonion torus (or Cayley torus) appears as a coordinate algebra of extended affine Lie algebras of type A2 and F4. A generalized version of the octonion torus, called division -graded alternative algebras, is classified in this paper. Using the result, we can complete the classification of division -graded Lie algebras, up to central extensions, which are a generalization of the cores of extended affine Lie algebras of type A2.