3-Engel and (2,5)-torsion-free implies class four for groups

Statement
Suppose $$G$$ is a 3-Engel group. Suppose, further, that $$G$$ does not have any 2-torsion or 5-torsion, i.e., $$G$$ does not have any non-identity element of order 2 or 5. Then, $$G$$ is a group of nilpotency class four: it is a nilpotent group and its nilpotency class is at most four.

Similar facts for 3-Engel groups

 * Equivalence of definitions of 3-Engel group: Shows that the 3-Engel condition is equivalent to Levi class two, i.e., the normal closure of every element having class at most two.
 * 3-Engel implies locally nilpotent

Similar facts for other Engel groups

 * 2-Engel implies class three for groups
 * 2-Engel and 3-torsion-free implies class two for groups
 * 4-Engel and (2,3,5)-torsion-free implies class seven for groups