Thanks very much for your comments.
The starter will turn the engine, it just doesn't fire when the immobiliser is set.
I can be a bit more precise in my description of ther problem. If the ignition is turned off, and the engine immediately restarted, there is no problem. There seems to be a delay of about 10 seconds, and then the direction indicators flash once, and the immobiliser is set. If then attempt to restart, the red and green lights in the mirror flash alternately, as you describe. The car then has to be locked and unlocked with the remote control before the engine will fire.
This does seem strange, and is not in accordance with the manual.
However, if I simply unlock the car from outside, get in, and start the engine immediately, there is no problem.