Free Lie ring on an abelian group

Definition
Let $$A$$ be an abelian group. The free Lie ring on $$A$$ is defined as a Lie ring $$\mathcal{L}(A)$$ along with an embedding of $$A$$ as a subgroup of the additive group of $$\mathcal{L}(A)$$ such that, for any Lie ring $$M$$ and any group homomorphism $$\varphi$$ from $$A$$ to $$M$$ $$\varphi$$ extends uniquely to a homomorphism of Lie rings from $$\mathcal{L}(A)$$ to $$M$$.

Note that this notion is quite different from that of the associated Lie ring of a group. The associated Lie ring, when applied to an abelian group $$A$$, would give an abelian Lie ring whose additive group is precisely $$A$$. The free Lie ring, on the other hand, would have separate elements for various brackets of things in $$A$$, set to be equal only based on the rules of addition in $$A$$ and the identities that Lie rings must satisfy.