The shear-induced states of orientation of the lamellar lyotropic mesophase of tetraethylene glycol monododecyl ether (C12E4) in water are investigated by polarizing microscopy, viscometry, small-angle light scattering, and deuteron NMR spectroscopy. The solution containing 40 w/w % surfactant shows a continuous transition from a state of aligned layers, whose normal is parallel to the velocity gradient, to a close packing of multilamellar vesicles (onion state). The size of the vesicles (diameter d) is controlled by the shear rate (gamma) over dot, following the relationship d proportional to (gamma)over dot(a) with a approximate to -0.5.