Lazard-divided Lie operad for a set of primes

Definition
Suppose $$\pi$$ is a set of primes. The $$\pi$$-Lazard-divided Lie operad is an operad over $$\mathbb{Z}$$ that is intermediate between the Lie operad over $$\mathbb{Z}$$ and the Lie operad over $$\mathbb{Q}$$. It is defined as follows: consider the Lie operad over $$\mathbb{Q}$$, and take the $$\mathbb{Z}$$-suboperad generated by the following operations, one for each prime number $$p \in \pi$$:

$$(x_1,x_2,\dots,x_p) \mapsto \frac{1}{p}[[ \dots [x_1,x_2],\dots,x_{p-1}],x_p]$$

Particular cases

 * If we take $$\pi$$ to be the empty set of primes, the $$\pi$$-Lazard-divided Lie operad is the usual Lie operad over the ring of integers.
 * If we take $$\pi$$ to be the set of all primes, the $$\pi$$-Lazard-divided Lie operad is the usual (with no prefix) Lazard-divided Lie operad.
 * The case where $$\pi$$ is an initial segment of the set of primes is of particular interest.