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16 gauge should be fine. i would worry more about if the insulation of the speaker wire could handle the heat if anything. but assuming 700ma at 12V would produce 8.4 watts of heat, correct me if im wrong please. so it should be fine considering the short amount of time they will run.
8.4W is correct, but it's not 8.4W of heat, and not 8.4W of heat in the wire. Some of it is lost in the wire as heat, some in the ballast as heat, some heat from the tubes, and some visible light energy.

I used a 1A fuse in my setup in case the wire was cut somehow and shorted or there was some malfunction in the lights or ballast. The fuse holder was less than $1 and the fuses were $1 or so for 4 pieces. Cheap insurance. I put it there to protect against frayed, damaged wiring, or some other malfunction that causes excess current draw creating a potential fire hazard.
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