Did anybody study those subcategories of triangulated categories that are closed with respect to "extensions" (in the sense of distinguished triangles; in particular, any such $B$ is additive)? If we consider these "extensions" as short exact sequences in $B$, we "almost" obtain the structure of a Quillen's exact category on it. The problem is that the corresponding admissible monomorphism are only "weak kernels" in $B$ in general (dually, admissible epimorphisms are only weak cokernels). Does some part of the theory of exact categories can be applied to this setting yet; can one associate an abelian category to $B$ (endowed with this "almost exact structure") in a reasonable way? Are there any results known for "almost exact" categories in this sense?
Upd. Possibly, it is better to consider a smaller set of exact sequences in this setting (so that the corresponding kernels and cokernels will be "strong").