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.