whenever (and thats 1-2 times / year) I book a paid business-class Star-rtw ticket (bought and ticketed by LH in ZRH) I am booked on inner NorthAmerican two-calss flights with UA in booking-class 'F'.

Later, for long-haule transcontinental flights, I will check-in very early and try to change that segment to a same-day.three-class flight. UA then automatically (if space is available) will give me an 'F' seat (on this now three-class flight).