Graded polynomial identities and Specht property of the Lie algebra sl2sl2

Let G   be a group. The Lie algebra sl2sl2 of 2×22×2 traceless matrices over a field KK can be endowed up to isomorphism, with three distinct non-trivial G  -gradings induced by the groups Z2Z2, Z2×Z2Z2×Z2 and ZZ. It has been recently shown (Koshlukov, 2008 [8]) that for each grading the ideal of G-graded identities has a finite basis.In this paper we prove that when char(K)=0char(K)=0, the algebra sl2sl2 endowed with each of the above three gradings has an ideal of graded identities IdG(sl2)IdG(sl2) satisfying the Specht property, i.e., every ideal of graded identities containing IdG(sl2)IdG(sl2) is finitely based.