Inner automorphism of a Lie ring

Definition

Suppose $L$ is a Lie ring. We call an automorphism $\sigma$ of $L$ a basic inner automorphism of $L$ if $\sigma =\exp(d)$ where $d$ is an inner derivation of $L$ (i.e., the adjoint action by some element) that is also an exponentiable derivation.

We call an automorphism of $L$ an inner automorphism if it can be expressed as a product of basic inner automorphisms. Note that the inverse of a basic inner automorphism is basic inner, so we see that the inner automorphisms are precisely the automorphisms in the subgroup generated by the basic inner automorphisms.