On constructing weight structures and extending them to idempotent extensions

We describe a new method for constructing a weight structure $w$ on a triangulated category $C$. For a given $C$ and $w$ it allow us to give a fairly comprehensive (and new) description of those triangulated categories consisting of retracts of objects of $C$ (i.e., of subcategories of the Karoubi envelope of $C$ that contain $C$; we call them idempotent extensions of $C$) such that $w$ extends to them. In particular, any bounded above or below $w$ extends to any idempotent extension of $C$. We also discuss the applications of our results to certain triangulated categories of ("relative") motives.