Nilpotency class of associated Lie ring equals nilpotency class of quotient of group by nilpotent residual

Brief version
The nilpotency class of the associated Lie ring of a group equals the nilpotency class of the quotient group of the group by its nilpotent residual.

Detailed version
Below, $$G$$ is an arbitrary group and $$L(G)$$ is its associated Lie ring. The statement is that in both cases below, the "Formulation for $$L(G)$$" is equivalent to the "Formulation for $$G$$."

Facts used

 * 1) uses::Explicit description of lower central series of associated Lie ring of a group