Knixon

2007 IS 250 - after being jumped off the vehicle runs while being connected to the other vehicle. When disconnected from the jump vehicle, it runs for a few minutes, then a lot of lights flash on the dash and the car shuts off. If I repeat the process and put the car in gear, the car shuts down once placed in gear. I have only left the Lexus connected to the jump vehicle for a couple of minutes.

PFB

Chances are your own battery is defective, or neeeds a proper full recharge.

Also check, or have someone check your alternator as well, but start with your battery.

BDSL

Sounds like alternator is bad.
With a dead battery, the engine should be able to run continuously off the alternator.