Another thought, just in case you're not aware of the immobiliser on these.
When I've had the immobiliser kick in, the engine has actually fired up and run for 1 - 2 seconds then its killed. I think your engine is running for longer than this before it dies, so I dont think its the immobiliser, but worth mentioning maybe to make you aware.
If / When it dies next time, check the dash lights to make sure you havent got the immobiliser symbol (key) on there.
The immobiliser on these seems to be particularly sensitive to battery voltage. You'd think if the battery had enough power to fire up the engine the immobiliser would always be happy, but from what I've seen thats not always the case.
If you do suspect an immobiliser problem, double check your batteries (car batteries, not the ones in your keyfob - theyre just used for the remote buttons not the immobiliser) are charged OK and maybe start using a different key for a while if you have one - see if that makes any difference!