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Yes, you can attach a cigar lighter plug to the Garmin red/black wires. Don't cut out the wire lump (yours looks like a square one, most are round). One issue I have had with this setup is that the cigar plug does not have any kind of retaining system and will tend to back out with vibration. I have had much more success with the BMW style plug and socket.
Hi,I've had success with Eastern Beaver for some of my accessory wiring projects.https://www.easternbeaver.com/Main/main.html