Abstract: A double-lift weave system comprising a plurality of hooks, each of which is provided with two arms mounted on a base and each of which includes a lift lip adapted to cooperate with one of the knives of two griffe frames moving in phase opposition in a reciprocating movement. The position of each lip is controlled in order to be selectively moveable with respect to a knife by an actuator disposed between the two arms. Each hook further includes two open shed heels which are controlled by the actuator and adapted to cooperate with open shed bars so as to be moveable with respect to the bars. The system is particularly applicable to weaving looms.