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 $\mathcal{M}_k$ over a field $k$, known as Grothendieck-Chow-Motives is based on choosing rational equivalence for $\sim$. But one can also choose other equivalence relations for $\sim$, and thus get a motivic category $\mathcal{M}^\sim_k$ , which might have different properties (being Tannakian for example).
Assume we have a motivic decomposition of a smooth, projective variety $X$ over $k$,
$M(X) = \bigoplus_{i\in I} M_i$,
in $\mathcal{M}_k$.
1.Is it known how the decomposition of $X$ in $\mathcal{M}^\sim_k$ 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 $\sim$ 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.
