Lie ring of derivations

Definition

Let $L$ be a Lie ring. The Lie ring of derivations of $L$, denoted $\operatorname {Der} (L)$, is defined as a Lie ring whose elements are the derivations of $L$, where:

• The additive structure is given by pointwise addition. Thus, the zero of this ring is the zero derivation.
• The Lie bracket is given by the commutator. Thus, if $d_{1},d_{2}:L\to L$ are derivations, their Lie bracket is defined as:

$[d_{1},d_{2}]:=d_{1}\circ d_{2}-d_{2}\circ d_{1}$.

In other words, $[d_{1},d_{2}](x)=d_{1}(d_{2}(x))-d_{2}(d_{1}(x))$.