E-engelian abelian groups of step $\le2$

It is proved that periodicity of the automorphism group or weak transitivity of an E-engelian torsion free group of step $\le2$ implies commutativity of its endomorphism ring. Some properties of the engelian ring of step $2$ are established. It is also shown that the endomorphism ring of a weakly transitive torsion free group is semiprime.