Lie ring satisfying Engel and bi-Engel degree constraints in terms of powering threshold and torsion-free threshold

Definition
A Lie ring satisfying Engel and bi-Engel degree constraints in terms of powering threshold and torsion-free threshold is a Lie ring satisfying the following two conditions:


 * 1) It is a bounded Engel Lie ring and its Engel degree is at most one more than its powering threshold.
 * 2) It is a bounded bi-Engel Lie ring and its bi-Engel degree is at most one more than its torsion-free threshold.