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.