Purple wire off the glow plug controller with the special connector on it powers the light. I would unhook the wire when this happens to see if the light goes out and stays out. If unplugging it stops the light from coming on focus on the connections to the glow plug controller and the controller itself.
Check the wiring to and from glow plug controller. If the wiring to it goes on and off (Wiring issue, bad ground, bad ignition switch, etc.) it will keep resetting the controller and it will go on for another 5-10 seconds. If the light goes on but does not turn the relay on and sends power to the glow plugs I would replace the glow plug controller.
To see if it is powering the glow plugs put a test light or volt meter on the big terminal of the glow plug controller with the bent metal part on it. If that side is powered then the glow plugs will be on. I would not unplug a glow plug to test as if the system senses 2-3 bad glow plugs it will shut down. So if you have a couple bad glow plugs and unplug a third it will shut off and your test will not show a true result.
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