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, $$[d_1,d_2](x) = d_1(d_2(x)) - d_2(d_1(x))$$.