My 07 JK had an intermittent battery light, it would come on during accelerating from about 45-50 and stay on while driving.
Stop the engine, then start again the light would be off, but come on again while driving.
I've checked the battery and it is good 13.9 VDC at rest, passes a high amp discharge test, and recharges normally, even with the battery light on.
Checked the alternator by disconnecting the main wire and it produces 14.3 volts at idle, RPM variation makes no difference.
Changed the ECU Engine control module, to a rebuilt unit, did not fix the problem.
removed new ECU replaced with old unit, no change.
Removed and cleaned all wire connections at battery, and alternator. now the light is always on.