Inner derivation from Lazard ideal is exponentiable

Statement
Suppose $$L$$ is a Lie ring and $$I$$ is a Lazard ideal in $$L$$. Then, for any $$u \in I$$, the inner derivation of $$L$$ arising via the adjoint action of $$u$$ is an exponentiable derivation of $$L$$.

Proof
Given: Lie ring $$L$$, ideal $$I$$ of $$L$$. A natural number $$c$$ such that the 3-local nilpotency class of $$I$$ is at most $$c$$ and $$I$$ is powered over all primes less than or equal to $$c$$. An element $$u \in I$$.

To prove: The adjoint map $$\operatorname{ad} u = x \mapsto [u,x]$$ is an exponentiable derivation of $$L$$.

Proof: The current version of the proof is incomplete and inaccurate, some fixing needs to be done