Relations between the associated Lie algebra and the adjoint group of a radical ring

On the other hand, one obtains a semigroup by considering the quasimultiplication a*b := a + b + ab. A ring is called radical, when this semigroup is actually a group, the "adjoint group".
One is interested in relations between these two structures associated with the ring. In both of them one can consider the property "metabelean", which means that commutators commute.
The main emphasis will be on a theorem which states that a radical ring is Lie metabelian if and only if its adjoint group is metabelean. The theorem thus furnishes a connection between the theory of Lie algebras and the theory of groups.