Triangulated subcategories of extensions, stable t-structures, and triangles of recollements

In a triangulated category TT with a pair of triangulated subcategories XX and YY, one may consider the subcategory of extensions X⁎YX⁎Y.We give conditions for X⁎YX⁎Y to be triangulated and use them to provide tools for constructing stable t-structures. In particular, we show how to construct so-called triangles of recollements, that is, triples of stable t-structures of the form (X,Y)(X,Y), (Y,Z)(Y,Z), (Z,X)(Z,X). We easily recover some triangles of recollements known from the literature.