As Team GB won another slightly improbable gold in the individual dressage, the BBC commentator announced that he would join in singing the national anthem. After Mo Farah had won the 10,000 metres, the BBC highlights programme the following morning cheekily ran footage shot inside the BBC studio during the event, showing their three athletics analysts jumping and cheering as Farah sped to the finish line. That evening, the BBC re-broadcast that footage. Afterwards, the studio analysts grinned in mock embarrassment, and presenter John Inverdale apologised for the temporary loss of impartiality.

During the following week, there were stories - disputed - that the BBC's Director-General was worried about the degree of cheerleading by the BBC of British competitors. But on the Saturday night, after Mo Farah had raced to a second gold, the BBC producer boldly and immediately presented backstage footage, not just of the studio analysts again jumping for joy, but of the race commentator, Steve Cram, rising from his seat to celebrate the double triumph. No apology this time for abandoning impartiality.

Yet how could a "British" broadcaster, funded by 94% of all British households, passionately supporting "their" athletes, not respond to that swelling sense of pride, as the medal rush built up momentum? The BBC executive master-minding the Olympics coverage, Roger Mosey, on The Media Show, drew a distinction between sports programmes - where a degree of cheerleading was perfectly acceptable - and news programmes, which needed to be objective.

Yet BBC news programmes, whilst quieter in tone, still disproportionately concentrated on British success. David Rudisha's astonishing 800 metres triumph - leading throughout, breaking the world record, and dragging all the finalists to personal bests, such that even the British runner finishing last recorded a time that would have won gold at the three previous Olympics - was swamped by news of gold medals the same day for three female Brits, in much lower profile events. 

There was another dimension of the patriotism debate (which brought the BBC perhaps unwelcome support from the Daily Mail) which has come to the fore: what is Team GB anyway?

One aspect of the potted history in Danny Boyle's quirky opening ceremony drew little attention: the careful segues, musical and pictorial, illustrating the four segments of the United Kingdom - England, Scotland and Wales (the constituents of Great Britain), and Northern Ireland.

Curiously, clips from rugby football matches were deemed the best way to capture the differences within the UK. As it happens, rugby is not yet even an Olympic sport. Moreover, Northern Ireland combines with Eire in rugby, to form the Ireland team. The four rugby teams periodically form a British Lions touring team, which includes non-British Irishmen: this team represents a geographical entity that has no legal status - the British Isles.

Just as people born in Northern Ireland can choose UK or Irish citizenship (or both), so NI athletes were able to choose between the GB and Irish teams. A majority chose Ireland, not least because one of the best represented sports was boxing, which - like rugby - is organised on an all-Ireland basis. That a Belfast boxer won a bronze medal for Ireland is just one of the quirks of the situation.

To add to the puzzlement, the UK team is tagged "Great Britain" - or GBR - in Olympic parlance: something of an embarrassment when three Northern Irish oarsmen won medals in the space of 24 hours. There was a long debate between the four "home country" football associations before it was reluctantly agreed to submit a "GB" men's football team (as it happens, with no players from Scotland or Northern Ireland). This scratch team actually progressed to the quarter finals before succumbing to the English habit of losing a knock-out tie on penalties. There will be no repeat of this experiment at Rio, as the four "home" associations have decided they do not wish to risk their individual access to the World Cup and European championship.

The mild illogicality of a nation that only issues UK passports being described as GB by the International Olympic Committee is just one of the oddities of the national question. Another is that Alex Salmond hopes that by 2016 Scotland will have its own team at the Rio Olympics - a prospect strongly opposed by the likes of Gordon Brown and Alastair Darling, not to mention top Scottish cyclist, Sir Chris Hoy.

Hoy's objection carries weight: the international cycling authorities decided to ban more than one entry per team for individual events, after the runaway GB successes in Beijing, which cost Hoy the chance to defend his individual sprint title (the slot was filled by an Englishman). A separate Scottish team would have allowed him an entry route. But he would not have been able to benefit from the huge investment in cycling training by UK Sport, and the pursuit team without Hoy might not have won its event. Even if Scotland obtained separate IOC status in 2015, the chances of assembling a meaningful Olympic challenge for 2016 would be slim.

The closing ceremony included film of John Lennon singing his emotional anthem, Imagine; reprised by the BBC in its closing credits sequence, in a specially performed version by Emile Sande. "Imagine there are no countries", invites Lennon. Yet in the Olympics, although the majority of events are for individuals, all competitions are reported as national outcomes. This is not surprising. The IOC is entirely organised on national lines. Indeed, Coubertin's primary motive in reviving the Olympics was to improve national character - France's defeat in the 1870 war with Prussia and Coubertin's admiration of British sporting manhood were key factors. Just one athlete was allowed to enter the London Olympics on a personal basis: a marathon runner from South Sudan, whose country has yet to arrange Olympics status with the IOC.

The appeal of national success and identity clearly still works. Nearly 10% of GB medal winners in 2012 were born abroad but wear their British nationality with pride. Andy Murray - unable to win a grand slam tennis tournament in his own right - somehow managed to destroy the world's top two players to win gold representing GB. He explicitly cited the inspiration he derived from watching on TV as three British athletes triumphed the night before his tennis final. One of those who inspired this Scot was a Muslim who had left behind his native Somalia at the age of 8.

Is national pride an admirable sentiment? And is it an effective motivator? There is clearly a "home" advantage evident in the recent medal performance of host countries - though part of this may simply be the product of being allowed to enter every event (GB had 75% more competitors in London than went to Beijing, but captured only 40% more medals). 

However achieved, such success creates a virtuous circle of vocal public support and enhanced performance. In turn, that underpins further funding of sports, at the individual and organisational level. Much of this funding comes through the National Lottery - a poor-value opportunity for gambling by the under-informed that has to be credited to the underwhelming Tory Prime Minister, John Major. 

Also key to this process is the state broadcaster, the BBC. Interestingly, Gordon Brown cited the BBC as another valuable product of union, whose status might be lost if Scotland became independent. That the BBC arguably under-serves Scotland is a case which has been strengthened by the government having forced the BBC to take over the Welsh language broadcaster, S4C, at a cost to the licence fee almost identical to the price tag attached to a Scottish broadcasting service (which the BBC refuses to countenance).

The relationship between the centre and the periphery has long been a sensitive one for the BBC, with command firmly located in London, however well-established the BBC's presence in national and regional strongholds (think in terms of Edward I's castles). More than 80 years ago, Reith had inscribed in the portals of Broadcasting House - long before the World Service was even thought of - the motto "nation shall speak peace unto nation". Perhaps he was thinking even then of how to deal with Scotland, whilst insisting on a unitary structure.

In the event, the BBC won wide praise for its Olympics programming, even though, as it happens, the opening and closing ceremonies - and most of the actual television coverage of events - were not originated by the BBC, but by the Olympics organisation. Yet it was BBC commentators who substantially framed the Olympics for UK viewers, on the main BBC channels. And for the first time, the BBC managed to provide complete coverage of every event, in real time, through its website and through multi-channel platforms such as Sky.

This gave the BBC many bites at the audience cherry, with the result that viewing figures outstripped even the recent Jubilee levels: over 90% of the UK population watched at least 15 minutes of BBC output during the Games. The BBC's many expert commentators were reliable guides to events, even when sometimes shouting themselves hoarse as GB athletes approached the winning line. 

Amusingly, an ITV documentary on "The Real Chariots of Fire", shown just before the Olympics, included newsreel of Harold Abrahams - winner of the 100 metres in 1924 - commentating for the BBC and cheering home the winner of the 1500 metres in 1936. "Hooray," he exclaimed repeatedly, as the world record was broken by Jack Lovelock - a New Zealander!

Yet amongst the celebratory coverage could be found some serious journalism, explaining the detailed background of many a sport and competitor. There were repeat screenings of a mini-documentary presented by the estimable Clare Balding (one of the few out lesbians in British broadcasting) telling the shameful story of how slowly and grudgingly women were allowed into the Olympic arena. Nor was it just the crusty misogynists of the IOC who were to blame. How many of us were aware that three weeks after Roger Bannister achieved immortality by running the first sub-four minute mile, Diane Leather became the first woman to run a mile in under five minutes?

Much of the credit for the detail and careful positioning of the BBC coverage can be attributed to Mosey, who has been rewarded with temporary appointment to the top position in BBC Television, taking over from the newly-announced Director-General. Mosey embodies that shrewd mixture of political nous and professional expertise that thrives inside the BBC - exactly the combination which protects the BBC's unique position in the UK.

The opening ceremony and the parentage of Team GB's most popular gold medallist (Jessica Ennis) underlined how ethnicity can no longer be the defining characteristic of Britishness. In Brixton, streets were decorated with Jamaican flags the day after Usain Bolt's 100 metres victory - though marking Jamaica's 50th anniversary of independence was the ostensible reason for the display.

Perhaps there are other, subtler, forms of affinity at work. Training together clearly instils an extra layer of loyalty and confidence. Bolt and his training partner, Yohan Blake, took gold and silver in the 100 and 200 metres. Two pairs of British canoeists did the same in their event, after years of training together. Mo Farah's training partner - the American Galen Rupp - was predicted by their coach as being just a step behind Mo in ability, and duly took the silver in the 10,000 metres. The pair ran a lap of honour wrapped in their respective flags.

The British cyclist who won the individual sprint gold medal - who only secured his place in the event as the sole permitted GB contestant by out-performing the incumbent Olympic champion - said afterwards that he felt lonely in a solo event, away from the team with whom he had already won gold in the men's pursuit. The French silver medallist - beaten for the first time in eight attempts by this upstart - pointedly asked what had made the difference: team spirit and massive crowd support, came the answer.

The BBC was the megaphone which carried the fierce vocal support for Team GB at each venue to the watching millions. In adding its own cheers and editorial shaping, to enhance this effect, it attracted some purist criticism, but mostly warm praise. Commercial stations have effectively conceded primacy to the national broadcaster when it comes to national occasions. Like the monarchy - whose anniversaries it also comprehensively celebrates - the BBC embodies, and expresses, a national spirit. That such a spirit can be invoked, to such effect, at regular intervals, tells us something about ourselves - and the BBC - which should make us think long and hard.