Short $$(\textbf{SL}_2\times \textbf{SL}_2)$$-structures on Lie algebras

Abstract

Abstract$$\textbf{S}$$ 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 $$\textbf{S}$$ S -structures. Here we deal with a situation not covered by these gradings: the short $$(\textbf{SL}_2\times \textbf{SL}_2)$$ ( SL 2 × SL 2 ) -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.