Non-associative Gröbner bases, finitely-presented Lie rings and the Engel condition, II

We give an algorithm for constructing a basis and a multiplication table of a finite-dimensional finitely-presented Lie ring. Secondly, we give relations that are equivalent to the n-Engel condition, and only have to be checked for the elements of a basis of a Lie ring. We apply this to construct the freest t-generator Lie rings that satisfy the n-Engel condition, for (t,n)=(2,3),(3,3),(4,3),(2,4).