Derived subring of a Lie ring

Definition
Let $$L$$ be a Lie ring. The derived subring or commutator subring of $$L$$, denoted $$L'$$, or $$[L,L]$$ is defined in the following ways:


 * It is the additive subgroup generated by all elements of the form $$[x,y]$$, where $$x,y \in L$$
 * It is the Lie subring generated by all elements of the form $$[x,y]$$, where $$x,y \in L$$
 * It is the Lie ideal generated by all elements of the form $$[x,y]$$, where $$x,y \in L$$

In situations where the Lie ring is an algebra over some field or ring, the derived subring is also a subalgebra and an ideal over that field or ring. In those cases, it may be termed the derived subalgebra or commutator subalgebra.