Lie ring arising as the double of a 3-additive Lazard Lie cring

Definition
A Lie ring $$L$$ is termed a Lie ring arising as the double of a 3-additive Lazard Lie cring if there exists a 3-additive Lazard Lie cring with cring operation $$*$$, sharing the same underlying set and additive group as $$L$$, and such that:

$$[x,y] = 2(x * y) \ \forall \ x,y \in L$$