Let \(\rho _\ell \) be a semisimple \(\ell \) -adic representation of a number field K that is unramified almost everywhere. We introduce a new notion called weak abelian direct summands of \(\rho _\ell \) and completely characterize them, for example, if the algebraic monodromy of \(\rho _\ell \) is connected. If \(\rho _\ell \) is in addition E-rational for some number field E, we prove that the weak abelian direct summands are locally algebraic (and thus de Rham). We also show that the weak abelian parts of a connected semisimple Serre compatible system form again such a system. Using our results on weak abelian direct summands, when K is totally real and \(\rho _\ell \) is the three-dimensional \(\ell \) -adic representation attached to a regular algebraic cuspidal automorphic, not necessarily polarizable representation \(\pi \) of \(\textrm{GL}_3(\mathbb {A}_K)\) together with an isomorphism \(\mathbb {C}\simeq {\overline{\mathbb {Q}}}_\ell \) , we prove that \(\rho _\ell \) is irreducible. We deduce in this case also some \(\ell \) -adic Hodge theoretic properties of \(\rho _\ell \) if \(\ell \) belongs to a Dirichlet density one set of primes.
