  Let G be the pro-algebraic group attached to the tannakian category of
polarizable rational Hodge structures. We show that the quotient of G by its
derived group is the Serre group, the derived group of G is the simply
connected covering of the adjoint group of G, and that the adjoint group G is a
product of specific simple algebraic groups. As the Mumford--Tate groups are
exactly the algebraic quotients of G, this also describes them.
