Inner-Lazard Lie ring implies every inner derivation is exponentiable

Statement
Suppose $$L$$ is an inner-Lazard Lie ring, i.e., there exists a natural number $$c$$ such that the 3-local nilpotency class of $$L$$ is at most $$c$$ and $$L$$ is powered over all primes strictly less than $$c$$.

Then, every inner derivation of $$L$$ is an exponentiable derivation, i.e., for any $$u \in L$$, the adjoint map $$\operatorname{ad} u$$ is exponentiable.

Facts used

 * 1) uses::Exponential of derivation is automorphism under suitable nilpotency assumptions: We use the weaker version of the result.

Proof
Given: Inner-Lazard Lie ring $$L$$ and a natural number $$c$$ such that the 3-local nilpotency class of $$L$$ is at most $$c$$ and $$L$$ is powered over all primes strictly less than $$c$$. Element $$u \in L$$

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

Proof: