Minimal varieties of graded Lie algebras of exponential growth and the special Lie algebra sl2sl2

The Lie algebra sl2=sl2(K)sl2=sl2(K) of 2×22×2 traceless matrices over a field KK has only three non-trivial G-gradings when G   is a group, the ones induced by G=Z2G=Z2, Z2×Z2Z2×Z2 and ZZ. Here we prove that when char(K)=0char(K)=0, the variety varG(sl2)varG(sl2) of G  -graded Lie algebras generated by sl2sl2, is a minimal variety of exponential growth, and in case G=Z2×Z2G=Z2×Z2 or ZZ, varG(sl2)varG(sl2) has almost polynomial growth.