A Polynomial Upper Bound for the Nilpotency Classes of Engel‐3 Lie Algebras over a Field of Characteristic 2

A Lie algebra L is called an Engel-n Lie algebra if it satisfies the additional condition that ad(b) = 0 for all b. Engel Lie algebras are related to the ”restricted Burnside problem”. This can be stated as follows: Given integers n and r, is it true that there is an upper bound on the orders of finite rgenerator groups of exponent n? The answer to this question is yes. In 1959 P. Hall and G. Higman[1] made, given some assumptions about finite simple groups, the following reduction. ”It is sufficient to look at B(r, n) where n is a power of a prime.” Here B(r, n) is the (relatively) free r-generator group of exponent n. From the classification of finite simple groups we have that the assumptions of Hall and Higman are valid. The relationship with Lie algebras comes from the equivalence of the following two statements.