Bsm (second method) is not for suppling per se, it is for creating balance/reactions to aids. The first actions are standing flexions done in hand. First, lifting the bit lightly in the mouth to create mobility in the jaw and a better balance. The second action is millimeters of lateral flexibity (at the atlas/axis). The third action is to calculated to be able to mobilize the jaw and extend the neck. Then those actions are done in hand in walk. Then when the rider mounts the mouth is more educated as to the use of half halts (demi arrets) and helps with balance/ease of the horse choosing to go forward, etc. The horse is trained progressively (and not in few days). (Lateral flexions are very calculated..at the atlas-axis) and not focused on longitudinal flexion which comes over time and in motion).
Some information about how these actions cause the horse to use its body as a response: Philippe Karl Ecole de Légèreté - News: Taking over equestrian culture
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