  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.
