Centralizer-free ideal implies derivation-faithful

Statement
Suppose $$L$$ is a Lie ring and $$I$$ is a centralizer-free ideal of $$L$$. In other words, $$I$$ is an ideal of $$L$$ and the centralizer of $$I$$ in $$L$$ is the zero subring. Then, $$I$$ is a derivation-faithful Lie subring (and hence a fact about::derivation-faithful ideal) of $$L$$.

Analogues in groups

 * Normal and centralizer-free implies automorphism-faithful
 * Normal and self-centralizing implies coprime automorphism-faithful

Proof
Given: A Lie ring $$L$$, a centralizer-free ideal $$A$$ of $$L$$. A derivation $$d$$ of $$L$$ whose restriction to $$A$$ is the zero map.

To prove: $$d(l) = 0$$ for all $$l \in L$$.

Proof: We first prove that for any $$a \in A$$, $$[dl,a] = 0$$. For this, note that:

$$d([l,a]) = [dl,a] + [l,da]$$.

Since $$A$$ is an ideal and $$a \in A$$, $$[l,a] \in A$$, so since $$d$$ is zero on $$A$$, the left side is zero. Since $$d$$ is zero on $$A$$, $$da = 0$$, so $$[l,da] = 0$$. Thus, $$[dl,a] = 0$$.

Hence, $$dl \in C_L(A)$$. By assumption, $$C_L(A) = 0$$, so $$dl = 0$$.