Ideal property is centralizer-closed

Statement
Suppose $$L$$ is a Lie ring, $$S$$ is an ideal of $$S$$, and $$C = C_L(S)$$ is the centralizer of $$S$$ in $$L$$. Then, $$C$$ is also an ideal of $$S$$.

Analogue for group theory

 * Normality is centralizer-closed

Generalizations
The general version of this result is: invariance under any set of derivations is centralizer-closed, which reduces to this result when we take the inner derivations (see Lie ring acts as derivations by adjoint action). Another special case of this general result is:


 * Derivation-invariance is centralizer-closed

Hands-on proof using Jacobi identity
Given: A Lie ring $$L$$, an ideal $$S$$ of $$L$$. $$C = C_L(S)$$ is the centralizer of $$S$$ in $$L$$.

To prove: $$C$$ is an ideal of $$L$$, i.e., for any $$x \in L$$ and $$c \in C$$, $$[x,c] \in C$$.

Proof: To show $$[x,c] \in C$$ it suffices to show that $$[[x,c],s] = 0$$ for all $$s \in S$$.

By the Jacobi identity:

$$[[x,c],s] + [[c,s],x] + [[s,x],c] = 0$$.

Since $$C$$ centralizes $$S$$, $$[c,s] = 0$$, so the second term is zero. Further, since $$S$$ is an ideal, $$[s,x] \in S$$, and since $$C$$ centralizes $$S$$, $$[[s,x],c] = 0$$. Thus, both the second and third term on the left side are zero, forcing $$[[x,c],s] = 0$$.