Truncated exponential of derivation is automorphism under suitable nilpotency assumptions

Weaker version: global powering and torsion assumptions
Suppose $$R$$ is a non-associative ring (i.e., a not necessarily associative ring) and $$d$$ is a derivation of $$R$$ satisfying the following:


 * 1) The minimum of (the nilpotency of $$d$$) and (the weak binilpotency of $$d$$) is at most one more than the powering threshold for $$R$$.
 * 2) The weak binilpotency of $$d$$ is at most one more than the torsion-free threshold for $$R$$. In other words, there exists a natural number $$m$$ such that $$d^i(x) * d^j(y) = 0$$ for all $$x,y \in R, i + j \ge m$$, and $$R$$ is $$p$$-torsion-free for all primes $$p < m$$.

Then, define the truncated exponential:

$$\exp_r(d) := \operatorname{id} + \frac{d}{1!} + \frac{d^2}{2!} + \dots + \frac{d^r}{r!}$$

where $$r$$ is any number that is at least equal to ((minimum of the nilpotency of $$d$$ and the weak binilpotency of $$d$$) - 1) and at most equal to the powering threshold for $$R$$. We then have that:


 * $$\exp_r(d)$$ is an endomorphism of $$R$$ as a ring.
 * If we further assume that $$d$$ is nilpotent (regardless of the value of its nilpotency), then $$\exp_r(d)$$ is an automorphism.

Subcases
We denote by $$B$$ the weak binilpotency of $$d$$, by $$N$$ the nilpotency of $$d$$, by $$P$$ the powering threshold of $$R$$, and by $$T$$ the torsion-free threshold of $$R$$. Note that $$P \le T$$ by definition. We allow the values to be $$\infty$$ unless specified otherwise. We need $$\min \{ N, B \} \le 1 + P$$ and $$B \le 1 + T$$ for all cases. The cases are as follows:

Particular cases
We consider some Lie rings.

Facts used

 * 1) uses::Exponential of nilpotent derivation with divided Leibniz condition powers is endomorphism