  We introduce a tensor decomposition of the $\ell$-adic Tate module of an
abelian variety $A_0$ defined over a number field which is geometrically
isotypic. If $A_0$ is potentially of $\GL_2$-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.
