It is well known that there are several adequate equivalence relations for algebraic cycles (see https://en.wikipedia.org/wiki/Adequate_equivalence_relation for a list including definitions).
The category of motives Mk over a field k, known as Grothendieck-Chow-Motives is based on choosing rational equivalence for ∼. But one can also choose other equivalence relations for ∼, and thus get a motivic category M∼k , which might have different properties (being Tannakian for example).
Assume we have a motivic decomposition of a smooth, projective variety X over k,
1.Is it known how the decomposition of X in M∼k will differ from the above?
1.1.Is it true, that one will always have less or at most the same number of summands?
2.Can you give a specific example (aside from projective spaces)?
I am interested in the case for ∼ to be alg, but every example is welcome, considering how little is probably known.
To give more context. I am trying to find out more about criteria for the motive of a variety to be not decomposable. So lifting and descent properties are what i am really interested in.