Adjoint group of a radical ring is abelian iff the radical ring is commutative

Statement
Suppose $$N$$ is a radical ring and $$G = 1 + N$$ is its adjoint group. Then, $$N$$ is commutative if and only if $$G$$ is an abelian group.

In particular, this applies to the case where $$N$$ is an an algebra over a field and $$G$$ is the corresponding algebra group.