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In British political history only four Labour prime ministers have won general elections outright: Clement Attlee, Harold Wilson, Tony Blair and now Sir Keir Starmer. Only Mr Blair and Mr Starmer are still alive.
Tony Blair says "of course" he talks to his successor frequently. Contradicting a report that he has been texting Mr Starmer advising on the transition into government, Mr Blair insists that they only discuss things and that he does not offer advice directly.
He can say that again. In his new 328-page manual On Leadership, the former prime minister writes: "A leader needs to generate optimism. No one wants to get on a plane with a depressed pilot. You want the person in charge to look like they have something to live for. They need to exude confidence, not anxiety."
The current prime minister is certainly not listening to that what with his "things can only get worse…" pep talk to the nation.
The "…before they get better" coda to that message has yet to be elaborated on by the new government. Mr Starmer's optimism for the future seems to go no further than reversing damage allegedly done by the Tories.
Comparisons will inevitably be drawn between the fresh observations from the last Labour prime minister and the baby steps in the first few months of the new Labour government.
Mr Blair's ideas range far wider than Britain but he will also be aware that he is echoing Mr Starmer's words when he talks about "missions" for government and needing 10 years to turn a country around.
Mr Blair's re-entry into public debate is the most significant step so far in his bid for political rehabilitation.
'I would have stayed if I could'
He is backing up his book, subtitled Lessons For The 21st Century, with public events and prominently-placed media interviews. Aged 71, he is still vigorous and ready to put his thoughts into the national conversation.
Mr Blair admits "I would have stayed if I could" and that "nothing makes me wish I was back in government more" than "the 21st-century technological revolution".
Yet for all the former prime minister's intellectual restlessness, Mr Starmer need not fear a direct challenge to his authority. Mr Blair is explicit that he is not bidding to return to office.
He says he is merely offering his reflections on power, drawn from his decade in Number 10 and the 18 years since then advising "Leaders" - he always uses a capital 'L' - around the world. Whether people agree with him or not, that is a useful exercise.
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A pariah in his own party
Tony Blair never went away as far as the big businesses and international leaders consulting him since he left office are concerned.
At home, he was almost a non-person for some years. Not so long ago, he could barely walk down the street for fear of a passer-by attempting a citizens' arrest over alleged war crimes in Iraq and Afghanistan.
Under the leadership of Jeremy Corbyn, he was a pariah to his own party - parodied on placards as 'Tony B liar'. Blair even mused briefly in a newspaper interview about moving permanently to the US.
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Some of the generalised lessons he offers in On Leadership are drawn from his bitter experiences after 9/11.
He notes ruefully: "Where the big risk comes is when foreign policy is pursued inconsistently when events and issues are reacted to in an ad hoc way when an awareness of the importance of clarity and coherence is neglected."
Elsewhere he has admitted that the UK and the US did not adequately think through or respond to the consequences of their interventions in the Middle East.
But the Iraq invasion was in 2003. Blair has been out of active politics for 18 years. He published his memoir, Tony Blair: A Journey, in 2010 and donated all proceeds, more than £5m, to a charity for injured soldiers.
His new interventions seek to draw unspecific lessons for the future from his experiences rather than looking back in self-justification.
After a highly lucrative period as a global voice for hire, he has devoted himself for the past seven years on a not-for-profit basis to the Tony Blair Institute for Global Change.
The institute now employs over 1,000 people, working in close to 50 countries. It holds an annual Future of Britain conference in London.
There is no mention of Mr Starmer in the book. Gordon Brown, the rival who eventually forced Blair out, is referred to only once, ambiguously, in a chapter headed Avoid Paranoia.
"A vital cog in the government machine, [Brown] aspired to succeed me, and sooner rather than later. My folk understandably didn't like this. I used to say, 'he's entitled to want the job'," Blair writes nonchalantly.
Three stages in a leader's journey
Tony Blair reckons there are three stages in the "journey" of leaders, common to all countries and political systems.
Listening and learning in the first flush of power. Then "they think they know everything". And finally, maturity: the realisation that they don't know everything, and a renewed willingness to listen.
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Not every leader gets to that "sweet arrival at discernment" but Mr Blair believes he did and wants to help other leaders get there quicker.
"What matters is what works" is a familiar mantra from the New Labour years. In his book, Mr Blair says leaders should take responsibility, assemble the best teams, act quickly and decisively and monitor that they are delivering service and security to their people - "democracy or not", he says intriguingly.
Mr Blair's brisk pragmatism and apparent lack of concern for how leaders get to their exalted positions carry echoes of Machiavelli's leadership handbook, The Prince, but without the vindictiveness. In contrast, Mr Blair counsels avoiding making enemies, forgiveness, and not reading critical attacks.
He avoids specifics when he can but some of his comments on current issues during this relaunch will be controversial.
World would cope with Trump, part two
He believes that the world would cope with another Donald Trump presidency and, perhaps thinking of the impeachment of his friend Bill Clinton, he is "wary of weaponising the justice system politically… So, yes apply the law to leaders as to citizens, but there is no harm and much good in exercising judgement as to when doing so is truly necessary".
He bemoans the failure of subsequent UK governments to preserve the three central international alliances that he left in place, as America's most trusted ally, at the heart of Europe, and as a leader in development for poorer countries.
Brexit, he says, has adversely replaced short-stay young migrants from the EU with immigrants from former British colonies who bring their families.
Above all, Mr Blair is an evangelist for the opportunities presented by technology and artificial intelligence.
He has long argued that digital ID cards are the only way to curb illegal immigration. He was one of the international celebrities who appeared on platforms with the cryptocurrency king Sam Bankman Fried, who is now a convicted fraudster.
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Mr Blair expresses similar wide-eyed admiration for the possibilities opened up by big tech and its pioneers, including Elon Musk.
He lists many practical technological applications "by governments across the globe" now under way. With the arrival of AI he says they will change the nature of the state and urges to "get an understanding of this revolution" and adopt best practice.
Mr Blair goes further than the Starmer government in his open embrace of new technology.
In other areas, his ideas are not the policy priorities of the new government. If he makes it to stage three of the leader's journey, Mr Starmer may come to see the thoughts of Mr Blair as unhelpful in the short term but constructive in the long run.
Until then he would be wise to follow his predecessor's advice to "leave a little space for some light or laughter".