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The rule is very simple. Oneworld emerald and sapphire members, and passengers travelling on a oneworld airline in Business or First Class, are entitled to access ANY lounge OPERATED BY a oneworld airline PLUS whatever lounge that airline has nominated as their "default" lounge at that airport. However, where a oneworld airline merely contracts with a non-oneworld airline or ground agency to use their lounge, lounge eligible pax travelling on OTHER oneworld airlines CANNOT use that lounge.

So, in FRA (for the moment ignoring access difficulties created by different terminals and Schengen/non-Schengen), a lounge-eligible passenger travelling on BA is entitled to access the CX, IB or JL lounges, but is not entitled to access the AF lounge. However, a lounge-eligible passenger travelling on AY is entitled to access any of the CX, IB or JL lounges, and also the AF lounge (because AY has nominated the AF lounge to be its "default" lounge in FRA).