2-Engel not implies class two for groups

Statement
It is possible for a group to be a 2-Engel group but not a group of nilpotency class two.

Opposite facts

 * 2-Engel and 3-torsion-free implies class two for groups
 * 2-Engel implies class three for groups

Example
Note that any finite example must be a 3-group of nilpotency class three. The smallest example is a group of order $$3^7 = 2187$$. This is the Burnside group $$B(3,3)$$, defined as the quotient of free group:F3 by the relations that the cube of every element is the identity (this relation set is infinite, but we can get a finite presentation because the group does eventually turn out to be finit).