I am reading Beilison Ginsburg Schechtman's "Koszul duality". In the section 1.3, they introduced the notion filtered triangulated categories with only one example, considering an abelian category with filtration and take its corresponding derived category.
I would like to see other examples of filtered triangulated category other than that. Any references or explicit examples are welcome.
The construction you mentioned, of the filtered derived categories over an abelian one can be easily done in the settings of the derived category of all dg-modules over a dg-algebra (explicitly it appears in the appendix of https://arxiv.org/abs/1902.09441).
On the other side if $\mathcal T$ is a triangulated category which is the base of a strong, stable derivator $\mathrm D$, then the construction of a filtered triangulated category over $\mathcal T$ can be done inside the category ${\mathrm D}({\mathbb Z})$, where $\mathbb Z$ is the poset of integers. Despite the differences in details, this is the idea lying behind the constructions in https://arxiv.org/abs/1711.06331 and https://arxiv.org/abs/1807.01505. Finally note also that all triangulated categories encountered in practice are basis of triangulated derivators. They are known some examples of triangulated categories which are not of this type, but they are "pathological".
Sorry for the late answer!
