Free Centre-by-nilpotent-by-abelian Lie Rings

We study the free Lie ring of rank 2 in the variety of all centreby-nilpotent-by-abelian Lie rings of derived length 3. This is the quotient L/([γc(L′), L] + L′′′) with c > 2 where L is the free Lie ring of rank 2, γc(L′) is the c-th term of the lower central series of the derived ideal L′ of L, and L′′′ is the third term of the derived series of L. We show that the quotient γc(L′) + L′′′/[γc(L′), L] + L′′′ is a direct sum of a free abelian group and a torsion group of exponent c. We exhibit an explicit generating set for the torsion subgroup.