Does the immobilizer shut down spark or the fuel pump?
I recently had an issue where the motor would crank freely but would not start. I was away from home and didn't have a meter or any other tools, but I a passer by had a can of starting fluid. The motor coughed and sputtered on ether, but again, would not run.
An hour later it started up and has run fine since.
It never lit the CEL, but I hooked the car up to my generic OBD scanner and it found a stored code of 1530 which it identified as Camshaft Adjustment - Bank 1. That doesn't seem to be related.
I ordered a new relay, and I'm about to order a new fuel pump - unless someone has thoughts that it might be something else?