The left (right) triangulated structures of the stable categories

Beligiannis and Marmaridis [\emph{Comm. in Algebra,} 22(12)(1994), 5021-5036] constructed the left and right triangulated structures on the stable categories of additive categories induced from some homological finite subcategories. We extend their results to slightly more general settings. As an application of our results we give some new examples of stable categories which have left or right triangulated structures from abelian model categories. An interesting outcome is that we can describe the pretriangulated structures of the homotopy categories of abelian model categories via the ones of stable categories.