Transmission gear box (1) comprising, at least partially accommodated in the housing (1), an output shaft (6), a rotating drive member (5) mounted to rotate freely about the shaft (6), a system (2) for rotating the rotating drive member (5) in forward and reverse gear and, arranged between the shaft (6) and the rotating drive member (5), a clutch mechanism (3) having an engaged state and a disengaged state, wherein the shaft (6) is free to rotate in any of the directions of rotation thereof, the automatic clutch mechanism (3) being configured, in forward gear, to shift from the disengaged state to the engaged state by rotating in forward gear the motor member (5) when the speed of rotation of the rotating drive member (5) is higher than that of the shaft (6), and from the engaged state to the disengaged state by rotating in forward gear the shaft (6) when the speed of rotation of the shaft (6) is higher than the speed of rotation of the rotating drive member (5). The clutch mechanism (3, 4) is also automatic in reverse gear.