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- Short (SL2 × SL2)-structures on Lie algebrasPublication . Beites, Patrícia Damas ; Córdova Martínez, Alejandra Sarina; Cunha, Isabel; Elduque, AlbertoS-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 (SL2 × SL2)-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.