Frobenius condition on a pretriangulated category, and triangulation on the associated stable category

As shown by Happel, from any Frobenius exact category, we can construct a triangulated category as a stable category. On the other hand, it was shown by Iyama and Yoshino that if a pair of subcategories $\mathcal{D}\subseteq\mathcal{Z}$ in a triangulated category satisfies certain conditions (i.e., $(\mathcal{Z},\mathcal{Z})$ is a $\mathcal{D}$-mutation pair), then $\mathcal{Z}/\mathcal{D}$ becomes a triangulated category. In this article, we consider a simultaneous generalization of these two constructions.