We report on the shear-thickening transition observed in dilute aqueous solutions of cetyltrimethylammonium tosylate (CTAT) at concentrations \(\). We have re-examined the kinetics of the shear-thickening transition using start-up experiments at rates above the critical shear rate \(\). Using simple well-defined protocols, we have found that the transient mechanical response depends dramatically on the thermal and on the shear histories. Using the same protocols, flow birefringence experiments were carried out. The gap of a Couette cell containing the sheared solution has been visualized between crossed polarizers in steady shear conditions, as well as in start-up experiments. We show that the birefringent shear-induced phase starts from the inner cylinder and grows along the velocity gradient direction, as in a shear banding situation. However, around \(\) we have not observed a regime of phase coexistence (isotropic and birefringent).