S-structures on Lie algebras, introduced by Vinberg, represent a broad
generalization of the notion of gradings by abelian groups. Gradings by, not
necessarily reduced, root systems provide many examples of natural
S-structures. Here we deal with a situation not covered by these gradings: the
short (SL2xSL2)-structures, where the reductive group is the simplest
semisimple but not simple reductive group. The algebraic objects that
coordinatize these structures are the J-ternary algebras of Allison, endowed
with a nontrivial idempotent.Comment: 18 page