Logarithmable automorphism

Definition

Suppose $R$ is a non-associative ring and $\alpha$ is an automorphism of $R$. Suppose further that:

1. $\alpha -1$ is locally nilpotent, i.e., for every $x\in R$, there exists a natural number $n$, possibly dependent upon $x$, such that $(\alpha -1)^{n}(x)=0$.
2. $\alpha -1$ is an infinitely powered endomorphism of the additive structure of $R$, i.e., for all natural numbers $n$, $(\alpha -1)^{n}(R)$ is powered for all primes less than or equal to $n$.
3. The logarithm of $\alpha$, which can be defined because of (1) and (2), is a derivation of $R$.

We then say that $\alpha$ is a logarithmabl automorphism of $R$.