Rings of quasi-endomorphisms of some direct sums of torsion-free Abelian groups

We consider the a representation of quasi-endomorphisms of Abelian torsion-free groups of rank $4$ by matrices of order $4$ over the field of rational numbers $\mathbb{Q}$. We obtain a classification for quasi-endomorphism rings of Abelian torsion-free groups of rank $4$ quasi-decomposable into a direct sum of groups $A_1$, $A_2$ of rank $1$ and strongly indecomposable group $B$ of rank $2$ such that quasi-homomorphism groups $\mathbb {Q} \otimes \mathrm{Hom}(A_i, B)$ and $\mathbb {Q} \otimes \mathrm{Hom}(B, A_i)$ for any $i=1, 2$ have rank $1$ or are zero. Moreover, for algebras from the classification we present necessary and sufficient conditions for their realization as quasi-endomorphism rings of these groups.