Lie ring whose bracket is the double of a Lie bracket giving nilpotency class two

Definition
Suppose $$L$$ is a Lie ring with Lie bracket $$[, ]$$. We say that $$L$$ is the Lie ring whose bracket is the double of a Lie bracket giving nilpotency class two if there exists a Lie bracket $$\{, \}$$ with the same set $$L$$ and the same additive group structure, such that both these conditions are satisfied:


 * 1) $$2 \{ x, y \} = [x,y] \ \forall \ x,y \in L$$
 * 2) With the Lie bracket $$\{, \}$$, $$L$$ is a defining ingredient::Lie ring of nilpotency class two (i.e., its nilpotency class is at most two).

Note that this implies that $$L$$, with the original Lie bracket, is also of nilpotency class at most two.