Tate module tensor decompositions and the Sato-Tate conjecture for certain abelian varieties potentially of ${\mathrm{G}\mathrm{L}}_2$-type

We introduce a tensor decomposition of the $\mathrm{\ell }$-adic Tate module of an abelian variety $A_0$ defined over a number field which is geometrically isotypic. If $A_0$ is potentially of ${\text{GL}}_2$-type and defined over a totally real number field, we use this decomposition to describe its Sato–Tate group and to prove the Sato–Tate conjecture in certain cases.