Inner automorphism group of inner-Lazard Lie ring equals Lazard Lie group of its Lie ring of inner derivations

Statement
Suppose $$L$$ is an inner-Lazard Lie ring and $$Z(L)$$ is its center. The quotient $$L/Z(L)$$ is the Lie ring of inner derivations of $$L$$. We can also define the inner automorphism group of $$L$$.

The statement is that the inner automorphism group of $$L$$ can be canonically identified with the Lazard Lie group of the Lie ring of inner derivations.

Related facts

 * Inner-Lazard Lie ring implies every inner derivation is exponentiable: This is a preliminary to the proof.
 * Exponential of adjoint map is conjugation: This is sufficient for the proof in the case that $$L$$ is itself a Lazard Lie ring.