Lie ring graded over an abelian group

Definition
Suppose $$A$$ is an abelian group. A Lie ring graded over $$A$$, also called an $$A$$-graded Lie ring, is defined as a Lie ring $$L$$ equipped with the following additional data satisfying compatibility conditions:


 * Data: For every $$a \in A$$ an additive subgroup $$L_a$$ of $$L$$ such that the additive group of $$L$$ is the internal direct sum of $$L_a, a \in A$$. In other words, $$L = \bigoplus_{a \in A} L_a$$.
 * Compatibility conditions: For any $$a,b \in A$$ (possibly equal, possibly distinct), we have $$[L_a, L_b] \subseteq L_{a + b}$$.