I'd like some help understanding the idea of abelian-type Shimura varieties. In paricular, I understand an abelian-type Shimura datum $(G,X)$ generally parameterizes non-rational Hodge structures which therefore cannot be motivic, right? But by definition, the centrally isogenous connected Shimura datum $(G^{der},X^{der})$ is of Hodge type, so I guess the connected components of the Shimura variety $\text{Sh}_\Gamma(G,X)$ do have some interesting families of abelian varieties living over them, but maybe the association fails to be natural in some sense?
I'd like to understand how this plays out particularly in the example of a quaternion algebra $B$ over a totally real field $F$, split at exactly one infinite place. Then $\text{Sh}(B^\times, X)$ should be an algebraic curve; the family of Hodge structures on (say) the left action of $B^\times$ on $B$ has weight character $\mathbb{G}_m\to F^\times$ defined only over $F$ (and therefore does not give a rational Hodge structure), given by the identity inclusion at the split place, and the trivial character at every other place.
How can one see that the associated connected Shimura datum is Hodge type? Does this impyl that there is nevertheless some interesting abelian family over individual connected components of $\text{Sh}(B^\times, X)$? What is its relationship to the family of non-rational Hodge structures?
