Specht property for some varieties of Jordan algebras of almost polynomial growth

Let F be a field of characteristic zero. In [25] it was proved that U J2 , the Jordan algebra of 2 × 2 upper triangular matrices, can be endowed up to isomorphism with either the trivial grading or three distinct non-trivial Z2-gradings or by a Z2 × Z2-grading. In this paper we prove that the variety of Jordan algebras generated by UJ2 endowed with any G-grading has the Specht property, i.e., every TG-ideal containing the graded identities of UJ2 is finitely based. Moreover, we prove an analogue result about the ordinary identities of A1, a suitable infinitely generated metabelian Jordan algebra defined in [27].