Lie tori of type B2 and graded-simple Jordan structures covered by a triangle

We classify two classes of B2-graded Lie algebras which have a second compatible grading by an abelian group Λ: (a) Λ-graded-simple, Λ torsion-free and (b) division-Λ-graded. Our results describe the centreless cores of a class of affine reflection Lie algebras, hence apply in particular to the centreless cores of extended affine Lie algebras, the so-called Lie tori, for which we recover results of Allison, Gao and Faulkner. Our classification (b) extends a recent result of Benkart and Yoshii.Both classifications are consequences of a new description of Jordan algebras covered by a triangle, which correspond to these Lie algebras via the Tits–Kantor–Koecher construction. The Jordan algebra classifications follow from our results on graded-triangulated Jordan triple systems. They generalize work of McCrimmon and the first author as well as the Osborn–McCrimmon-Capacity-2-Theorem in the ungraded case.