  We show that the $\mathbb{Q}$-descents of the canonical
$\mathbb{R}$-variation of Hodge structure of Calabi-Yau type over a tube domain
of type $A$ can be realized as sub-variations of Hodge structure of certain
$\mathbb{Q}$-variations of Hodge structure which are naturally associated to
abelian varieties of (generalized) Weil type.
