Somewhat related to this thread but on a different tangent....
I remember seeing in the PBS/BBC documentary 'Airport' how Queen Elizabeth II flew down for a state visit to Australia and New Zealand... on British Airways!
Unlike Slick Willie's private airline (the 89th Airlift Wing) the British RAF does maintain a small fleet of VIP aircraft called the 'Queen's Flight.' Mostly VIP-corverted BAe RJs suited for short hops to Europe. The long-hauls are trusted to BA.
The entire first-class section was taken over and converted into a private suite for the Queen, while the business class and coach section were just your regular passenger flight from LHR to SYD. And HM traveled under an assumed name. Don't know if she took advantage of BA's mileage program...
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Causa latet vis est notissima - the cause is hidden, the results are well-known.