Characteristic implies powering-invariant in class two Lie ring whose torsion-free part is finitely generated as a module over the ring of integers localized at a set of primes

Statement
Suppose $$L$$ is a class two Lie ring. Denote by $$T$$ the torsion subgroup of the additive group of $$L$$. $$T$$ is a fully invariant subgroup of $$L$$, hence an ideal, hence the quotient $$L/T$$ is also a class two Lie ring. Suppose that $$L/T$$ is a Lie ring whose additive group is finitely generated as a module over the ring of integers localized at a set of primes. Explicitly, there exists a prime set $$\pi$$ such that $$L/T$$ is finitely generated as a $$\mathbb{Z}[\pi^{-1}]$$-module.

Suppose $$M$$ is a characteristic Lie subring of $$L$$. Then, $$M$$ is a powering-invariant Lie subring of $$L$$.