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2. Run the reaction motor. This rotates the Gyroscopes about their precession axis, accompanied by equal and opposite torque orthogonal to the spin axis. There can be no torque exerted by the reaction motor along a precession axis, therefore no substantial movement of the reaction flywheel.
It is not labeled, but I presume the reaction motor is that grey thing that spins up the red wheel. Also not labeled is whatever you consider the precession axis. There is the one main axle in the whole setup which is the rotation axis of the red wheel. There is no precession axissince there's only the motor producing spin and counterspin. So the red wheel spins one way, and the contraption to the left spins the other way. Still zero net angular momentum.