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

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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 D in a triangulated category satisfies certain conditions (i.e., (Z,Z) is a D-mutation pair), then Z/D becomes a triangulated category. In this article, we consider a simultaneous generalization of these two constructions.