Weak binilpotency

Definition
Suppose $$R$$ is a non-associative ring and $$\theta: R \to R$$ is an endomorphism of the additive group of $$R$$. Suppose $$n$$ is a positive integer. We say that $$\theta$$ is $$n$$-step-weak binilpotent if the following holds:

$$\theta^i(x) * \theta^j(y) = 0$$ for all $$x,y \in R$$ and all positive integers $$i,j$$ with $$i + j \ge n$$.

Note that if $$n > 1$$, then this is equivalent to checking that:

$$\theta^i(x) * \theta^j(y) = 0$$ for all $$x,y \in R$$ and all positive integers $$i,j$$ with $$i + j = n$$.

The weak binilpotency of $$\theta$$ is defined as the smallest $$n$$ for which $$\theta$$ is $$n$$-step-weak binilpotent.