Lie ring whose bracket is the multiple of a Lie bracket by the primorial of its 3-local nilpotency class

Definition
Suppose $$L$$ is a Lie ring with bracket $$[ \, \ ]$$. We say that $$L$$ is a Lie ring whose bracket is the multiple of a Lie bracket by the primorial of its nilpotency class if there is a nilpotent Lie ring structure $$ \{, \}$$ with the same underlying set and additive group as $$L$$ such that, if the 3-local nilpotency class with $$\{ \ , \ \}$$ is $$c$$, then:

$$\! [x,y] = m\{x,y \}$$

where $$m$$ is the primorial of $$c$$, i.e., the product of all primes less than or equal to $$c$$.