these symptoms suggest to me a failed ignition switch or failed control module.
the ignition switch is inexpensive so I would start there.
if the problem persists, then it is likely to be one of the vehicle's control modules which has failed and is playing up. To diagnose this, you could try unhooking modules one by one until you find the one that fixes the issue, or try having the vehicle scanned for faults using a BMW diagnostic computer.
again, I would start with the ignition switch
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