Constracting n-Engel Lie rings

A Lie ring L satisfies the n-Engel condition if [y,x,x,...,x]=0 (x occurs n times) for all x, y in L. By a result of Zelmanov every finitely generated Lie ring that satisfies the n-Engel identity is nilpotent, and hence finite-dimensional. We present methods for constructing a basis of the "freest" n-Engel Lie ring (over the integers) with t generators. We use Groebner basis type methods in free algebras, along with a linearisations of the n-Engel identity, valid over the integers.  Let L be generated as an abelian group by B={x_1,...,x_m}. Using a combinatorial approach, we have determined several sets of conditions on the elements of B only, that are necessary and sufficient for L to satisfy the n-Engel condition.