Maximal among abelian ideals implies self-centralizing in nilpotent Lie ring

Statement
Suppose $$L$$ is a fact about::nilpotent Lie ring and $$A$$ is maximal among abelian ideals in $$L$$: In other words, $$A$$ is an abelian ideal of $$L$$ not contained in any bigger abelian ideal of $$L$$. Then, $$A$$ is a self-centralizing Lie subring of $$L$$. In particular, $$A$$ is a self-centralizing ideal of $$L$$.

Analogues in other algebraic structures

 * Maximal among abelian normal implies self-centralizing in nilpotent (group)
 * Maximal among abelian normal implies self-centralizing in supersolvable (group)
 * Maximal among abelian normal not implies self-centralizing in solvable (group)