Nilpotent derivation with divided powers

Definition
Suppose $$R$$ is a non-associative ring (i.e., a not necessarily associative ring). A nilpotent derivation with divided powers for $$R$$ is a derivation with divided powers $$d^{(n)}, n \in \mathbb{N}_0$$, such that there exists a value $$r$$ called the nilpotency such that $$d^{(n)} = 0$$ for all $$n \ge r$$.

Note that for a nilpotent derivation with divided powers, it is definitely a necessary condition that the underlying derivation $$d^{(1)} = d$$ be a nilpotent derivation. However, this condition is not a sufficient condition. It is insufficient in two ways: first, it may well be the case for a nilpotent derivation that it cannot be extended to a derivation with divided powers. Second, it may be possible that the derivation can be extended but not to a nilpotent derivation with divided powers. Further, even if it can be extended to a nilpotent derivation with divided powers, the nilpotency may be larger.