Derived subring of uniquely p-divisible Lie ring is uniquely p-divisible

For a single prime
Suppose $$L$$ is a Lie ring and $$p$$ is a prime number such that $$L$$ is uniquely $$p$$-divisible, i.e., for every $$a \in L$$, there exists a unique $$b \in L$$ such that $$pb = a$$. Then, the fact about::derived subring of $$L$$, defined as the subring comprising all finite sums $$\sum_{i=1}^n[a_i,b_i]$$ with $$a_i,b_i \in L$$, is also uniquely $$p$$-divisible.

For a set of primes
Suppose $$\pi$$ is a set of prime numbers and $$L$$ is a Lie ring whose additive group is a group powered over $$\pi$$ (i.e., it is uniquely $$p$$-divisible for all $$p \in \pi$$). Then the derived subring of $$L$$ is also powered by $$\pi$$.

Generalizations

 * Derived series member of uniquely p-divisible Lie ring is uniquely p-divisible
 * Lower central series member of uniquely p-divisible Lie ring is uniquely p-divisible

Other similar facts

 * Center of uniquely p-divisible Lie ring is uniquely p-divisible
 * Upper central series member of uniquely p-divisible Lie ring is uniquely p-divisible

Opposite facts

 * Derived subgroup of uniquely p-divisible group need not be uniquely p-divisible
 * Derived subgroup of uniquely p-divisible class two group is uniquely p-divisible