Lazard correspondence between derived subgroup of free powered nilpotent group and derived subring of free powered nilpotent Lie ring

Statement
Suppose $$c$$ is a positive integer and $$\pi$$ is the set of all prime numbers less than or equal to $$c$$. Denote by $$G$$ the free pi-powered nilpotent group of class $$c$$ on a set $$S$$ and denote by $$L$$ the free pi-powered nilpotent Lie ring on $$S$$.

Note that if $$c$$ is not prime, then $$G$$ is a Lazard Lie group, $$L$$ is a Lazard Lie ring, and $$G$$ and $$L$$ are in Lazard correspondence. If $$c$$ is prime, however, neither $$G$$ nor $$L$$ is Lazard, and therefore they are not in Lazard correspondence.

However, the following is true regardless of whether $$c$$ is prime or not:

The derived subgroup $$[G,G]$$ is in Lazard correspondence with the derived subring $$[L,L]$$.