The method comprises a braiding step during which a braiding wire is wound in helical turns about, and along, a generatrix line which is closed in a ring about a main axis, so as to form a bead wire element called “braided torus” within which the braid wire has, on the one hand, a winding section which corresponds to the helical turns and which occupies a braid volume, and, on the other hand, in an extension of said winding section, a surplus length of braid wire called “excess section”. After the braiding step, there is a folding step during which the excess section is plastically deformed by folding, preferably by means of a punch and an anvil, so as to form an elbow which makes the excess section diverge with respect to the helical turns and makes said excess section protrude at least partly with respect to the braid volume.