Passing motivic decompositions from rational to algebraic equivalence

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.

Answer

Attribution
Source : Link , Question Author : nxir , Answer Author : Community