Not from a lighter socket.

Even 400 watts at full capacity is pushing the envelope. On the DC side, you're drawing nearly 30 amps. The wiring for a lighter just isn't designed for a sustained draw of that size.

The 400 watt inverters I've seen came with both lighter socket and battery clip attachments.

The 1000 watt I got has battery clips only. And the instructions strongly recommend a better (permanent) connection for anything other than casual use.