Rigid G2-representations and motives of type G2

We consider a family of motives associated to the rigid local system whose monodromy is dense in the simple algebraic group of type G2 and which has a local monodromy of order 7 at ∞. We prove an explicit Hilbert irreduciblity theorem for the associated étale realizations and deduce that the specialized motives at the points of irreducibility have motivic Galois group G2.