  Let $X$ be a complex smooth projective variety of dimension $d$. Under some
assumption on the cohomology of $X$, we construct mutually orthogonal
idempotents in $CH_d(X \times X) \otimes \Q$ whose action on algebraically
trivial cycles coincides with the Abel-Jacobi map. Such a construction
generalizes Murre's construction of the Albanese and Picard idempotents and
makes it possible to give new examples of varieties admitting a self-dual
Chow-K\"unneth decomposition satisfying the motivic Lefschetz conjecture as
well as new examples of varieties having a Kimura finite dimensional Chow
motive. For instance, we prove that fourfolds with Chow group of zero-cycles
supported on a curve (e.g. rationally connected fourfolds) have a self-dual
Chow-K\"unneth decomposition which satisfies the motivic Lefschetz conjecture
and consequently Grothendieck's standard conjectures. We also prove that
hypersurfaces of very low degree are Kimura finite dimensional.
