  Let K be a number field and {V_l} be a rational strictly compatible system of
semisimple Galois representations of K arising from geometry. Let G_l and
V_l^ab be respectively the algebraic monodromy group and the maximal abelian
subrepresentation of V_l for all l. We prove that the system {V_l^ab} is also a
rational strictly compatible system under some group theoretic conditions,
e.g., when G_l' is connected and satisfies Hypothesis A for some prime l'. As
an application, we prove that the Tate conjecture for abelian variety X/K is
independent of l if the algebraic monodromy groups of the Galois
representations of X satisfy the required conditions.
