BA has had a colourful recent history
They are two of global aviation's biggest birds - and by cosying up close to each other they hope to keep themselves out of the turbulence rocking the world's airlines, especially on key transatlantic routes.
British Airways and American Airlines are effectively applying for the right to break the US laws that prevent companies from co-operating too closely.
Anti-trust immunity, as it's known, is designed to stop cartels unfairly dominating markets and it is a big issue in aviation.
The reason is that of the three big airline alliances operating out of Heathrow Airport, two have anti-trust immunity.
Skyteam, including Delta, Northwest and Air France is one. The Star Alliance including United and US Airways is the other.
But One World - the British Airways and American Airlines alliance - is the third, with no such immunity.
If BA and AA can persuade the US Department of Transportation of their case they will be able to merge in all but name.
Incidentally they can't actually merge because of American laws preventing foreign ownership of US airlines.
But it would mean BA and AA could fix their fares, routes and schedules together.
A passenger using the BA website would also be able to buy through tickets on American to some of the smallest US destinations.
They would get BA air miles the whole way. Theoretically their bags would be checked all the way through, as if there was just one airline.
All of which may sound good for passengers.
But what happens when you let two massive airlines become practically one even more gigantic carrier?
Well this is what Sir Richard Branson - chief executive of BA's eternal rival Virgin Atlantic - says about it.
"Make no mistake. A BA/AA alliance would be bad for passengers, bad for competition and bad for the UK and US aviation industry".
Virgin believes that BA and AA are already too dominant - controlling 200,000 of the 480,000 take-off and landing slots at Heathrow.
And 62% of passengers travelling between Britain and America use one of the airlines.
This, Virgin says, is an advantage at Heathrow so big that the two shouldn't get anti-trust immunity even though their rivals have it.
For the transatlantic partners this tie-up has been on the wish-list for years.
They have already applied twice, in 1997 and 2001 but were told they would have to give up too many slots across the Atlantic to make it acceptable to them.
But now we have the Open Skies agreement which gives rights for a wider variety of airlines to fly between the US and the UK.
British Airways says this means that there is now more competition, and that BA and AA should be given a freer hand.
"I think it's a question of just how much is enough competition?", says aviation analyst Peter Morris of Ascend.
"If you look out of the UK you've already got 21 carriers operating scheduled services at the moment across the North Atlantic.
"I think BA would argue that by reducing the number and synchronising services they could provide a more efficient operation and reduce costs."
Regardless of the historical aspiration, the dire situation facing the world's airlines provides an added spur for BA and AA to do the deal.
Look at it from the American side.
"Few of us thought that we would ever look back fondly at $90 a barrel oil," American Airlines' chief executive Gerard Arpey told his shareholders in May.
"But, as you all know, the rise in the price of oil - and by extension jet fuel - has accelerated during the past several months to extraordinary levels.
"The slowing US economy and our industry's persistent overcapacity, have prevented us, or any other airline, from pricing our product in a way that reflects our skyrocketing costs."
No quick answer
That is the problem in a nutshell.
Its not quite as bad for British Airways, which made profits of nearly £900m last year despite an air crash and a PR disaster at Terminal Five.
But BA is facing fuel bills of up to £3bn this year.
Sir Richard Branson has already written to America's two presidential candidates, one of whom will end up making the decision, to persuade them to say no.
BA and AA have tried - and failed - twice before to have a tie-up.
The problem for BA and AA is that both Barak Obama and John McCain are pretty protectionist.
Will their desire to help a struggling American Airlines overcome their reluctance to give British Airways a foothold in a big American company?
The hiatus caused by the US election means the US authorities are unlikely to give a quick answer.
And even if they do give the go-ahead, the deal would certainly involve British Airways and American Airways giving up slots across the Atlantic.
But which slots?
Answering that question is likely to involve lawyers, regulators and a lot of wrangling.
Yet the fate of this proposed deal is worth watching closely.
The basic rule of the aviation market is that when the going gets tough, airlines either consolidate by merging or finding allies, radically cut costs, or go under.
BA and AA now wait to find out how far they can go in protecting themselves from an uncertain few years.