Jacobson's lemma for commuting outer inverses in rings

Let $a, b$ be elements in a ring $R$ with unity 1, and let $X_{1-ab}$ denote the set of commuting outer inverses of $1-ab$. In this paper, under the condition that $R$ is an abelian ring, we establish a bijection $\Omega: X_{1-ab} \rightarrow X_{1-ba}$, which is an isomorphism of posets with respect to Mitsch's partial order $\leq_{\mathcal{M}}$, and is a semigroup isomorphism with respect to the von Neumann product. It is also shown that there exists an isomorphism between certain subsets of $(X_{1-ab}, \leq_{\mathcal{M}})$ and $(X_{1-ba}, \leq_{\mathcal{M}})$ without any restriction on $R$.