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