Analogue of critical subgroup theorem for nilpotent Lie rings

Statement
Suppose $$L$$ is a nilpotent Lie ring of size $$p^n$$ where $$p$$ is a prime and $$n$$ is a natural number. Then, there exists a subring $$K$$ of $$L$$ that is both characteristic and derivation-invariant in $$L$$, and satisfies the following three conditions:


 * 1) $$[L,K] \le Z(K)$$.
 * 2) $$K/Z(K)$$ is an elementary abelian Lie ring.
 * 3) $$C_L(K) = Z(K)$$, i.e., $$K$$ is a self-centralizing Lie subring of $$L$$.