  We prove the Mumford--Tate conjecture for those abelian varieties over number
fields whose extensions to C have attached adjoint Shimura varieties that are
products of simple, adjoint Shimura varieties of certain Shimura types. In
particular, we prove the conjecture for the orthogonal case (i.e., for the
$B_n$ and $D_n^R$ Shimura types). As a main tool, we construct embeddings of
Shimura varieties (whose adjoints are) of prescribed abelian type into unitary
Shimura varieties of PEL type. These constructions implicitly classify the
adjoints of Shimura varieties of PEL type.
