By Nicholas Witchell
BBC Royal correspondent
We will never know what the Queen really thinks of George W Bush.
President Bush was coached on etiquette, US media said
Reputedly, when they met in 1991 during the Queen's visit to the White House of George Bush senior, George junior introduced himself to the Queen as the "black sheep of the family", and asked the Queen who was his equivalent in the Windsor family.
Sadly, the Queen's reply is not recorded, but young George's unconventional approach to the British monarch on that occasion was said to have earned him a very stern look from
his mother, the redoubtable Barbara Bush.
President George W Bush may have been reminded of that look when, on the south lawn of the White House on Monday, he fluffed his lines and suggested that the 81-year-old Queen has been on the throne since the 18th century.
Realising his mistake, and in front of all the cameras, George junior turned to the monarch and winked at her.
Now, the Queen is not used to being winked at. And certainly not by a fellow head of state, on a public platform in front of 7,000 people.
"She gave me a look that only a mother could give a child" quipped George, making as graceful a recovery as he could.
What, one wonders, was going through the presidential mind at that moment? Perhaps the Queen's expression did remind him of reproachful looks he received from his own mother, or other funny looks his idiosyncratic style may have earnt him over the years.
Ten presidents later she was still doing her bit for Britain even if that did mean turning a blind eye to George junior's strange little ways
Or maybe he was imagining the headlines which would greet his latest gaffe.
Because make no mistake, the US media has been full of stories about how the president was being coached on etiquette and protocol to try to prevent him placing one of his hand-tooled Texan cowboy boots in his mouth.
Those efforts failed. As the Washington Post reported the next morning, the White House's hopes of a gaffe-free visit were dashed within 14 minutes of George W Bush greeting the Queen.
Not, of course, that it mattered in the slightest. The Queen has witnessed far worse faux-pas over the 55 years of her reign to bat an eyelid at a mere Bush-ism.
She may well have been a little bemused by the wink, or had a quiet chuckle at the sight of "Dubya" looking uncomfortable in his white tie and tails, and she certainly looked a little flummoxed when she proposed a toast to the United States at the state dinner and found herself the only person standing up actually to drink the toast.
Part of job
But the thing about the Queen is that she has seen it all before. To her, it's all part of the job.
What will have mattered to her is the fact that this state visit - her fourth to the United States in the past 50 years - has clearly been a success.
The capital city of the world's superpower isn't an easy place to impress.
The Queen has impressed people in Washington
There is an understandable superiority about the city. Very, very few official visitors have the power to break through its studied indifference to just about everything that isn't American.
Queen Elizabeth II showed that she is one of those visitors who does have the presence to impress Washington's most powerful movers and shakers.
Partly it's because she's a major-league royal, which carries a mystique outranking any amount of "celebrity", and arouses a curiosity few be-suited politicians could muster.
Partly it's because of her venerable age and the fact that she has been a witness to, or a participant in, so many world events for so very long.
And partly it's because it was citizens from her country who, 400 years ago this month (as her visit recognised) established the first permanent non-native settlement in North America which, in turn, gave birth to the USA and a military alliance which has been tested in two world wars and various other conflicts.
Of course, that military alliance, which has been the bedrock of the so-called "special relationship", also gave rise to the visit's trickiest problem.
How should the Queen deal with the situation that the US and UK find themselves in Iraq?
The solution found by the Foreign Office, which designs the Queen's programme and writes all her speeches for visits such as this, was virtually to ignore it.
The Queen and Prince Philip attended a White House state dinner
The word "Iraq" was uttered only once by the Queen and then only in passing in the course of the four official speeches she delivered, which might be thought to be rather strange.
The fact that she could not refer more fully or directly to this shared endeavour between America and Britain does, of course, tell its own story.
If Iraq had been an unequivocal success, she would have talked about it in her speeches, and offered praise to those who had been involved - as she did to the elder George Bush when she visited Washington in 1991 after the first Gulf War.
No, in the context of a state visit which was intended to underline the success and importance of the relationship between London and Washington, Iraq was definitely not a subject to dwell on, or indeed to mention at all if it could possibly be avoided.
Dutiful as ever, the Queen did as she was asked by her government advisers. She kept her thoughts to herself, about Iraq, George W Bush and any other contradictions or oddities she may have encountered in the course of her six-day visit to America.
Above all whether you are a fan of monarchy or not it would, I suggest, be perverse not to agree that it was a personal triumph for an 81-year-old monarch who first visited Washington soon after the Second World War, when Harry Truman was president.
Ten presidents later she was still doing her bit for Britain even if that did mean turning a blind eye to George junior's strange little ways.