Centralizer-free ideal implies derivation-faithful

This article gives the statement and possibly, proof, of an implication relation between two Lie subring properties. That is, it states that every Lie subring satisfying the first Lie subring property (i.e., centralizer-free ideal) must also satisfy the second Lie subring property (i.e., derivation-faithful Lie subring)
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ANALOGY: This is an analogue in Lie rings of a fact encountered in group. The old fact is: normal and centralizer-free implies automorphism-faithful.
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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 Derivation-faithful ideal (?)) of $L$.

Related facts

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 $dl=0$.