Do any power inverter above 400 watt include a cigarette lighter adapter?
No, cigarette lighters cannot handle large loads, the highest model with a cigarette lighter is the PW400-12. If the cigarette lighter in your car is rated at 10 AMPS you will only be able to use the inverter up to 120 watts, 15 Amps will allow you to use it up to 180 watts and a 20 Amps up to 240 watts or up to 400 watts using the cables by connecting them directly to the battery.