David Cameron’s rendezvous with Nicolas Sarkozy at Lancaster House on Tuesday completes a whirlwind round of diplomacy intended to build an enhanced relationship with Paris and Berlin.

Mr Sarkozy and Mr Cameron will sign defence and nuclear treaties, but there are those in London who see the summit as part of a wider European power play.

The prime minister wants to strengthen ties with Berlin as well as Paris, creating an ad hoc version of the “trilateralism” that became a central – and ill-fated – tenet of Tony Blair’s early European policy.

Last weekend, Mr Cameron entertained Angela Merkel, German chancellor, at Chequers, exploiting the warming personal chemistry between the leaders and discussing a joint trade initiative.

“Angela Merkel loves Chequers,” confided one diplomat, who said it brought to mind one of the chancellor’s favourite television programmes: “The countryside reminds her of Midsomer Murders.”

London has long harboured hopes – perhaps fantasies – of breaking into the Franco-German relationship, that most recently bore fruit last month when Ms Merkel and Mr Sarkozy hammered out a deal on eurozone governance.

In the aftermath of the Iraq war in 2004, Tony Blair tried to link the EU’s three most powerful capitals in what the French call “a directoire”, an initiative that spluttered out amid dismal personal relations and a lack of a joint agenda.

“Tony Blair saw trilateralism as a political objective,” said one Number 10 insider. “David Cameron is much more practical. He sees this as more of an ad hoc arrangement – he will work with Paris and Berlin where it makes sense in the national interest.”

British officials say that, this time, the personal relationships are better. Mr Sarkozy is seen as mercurial but ready to do business, while Ms Merkel’s initially frosty relations with Mr Cameron have thawed.

Mr Cameron’s team believe there is also now more of a common agenda, including co-operation on defence, climate change, energy policy and Iran, while agreement may eventually be possible on EU economic reform.

But for Mr Cameron, the big prize is to persuade Paris and Berlin to back his plan to curb the EU budget.

Mr Sarkozy’s team like the idea of Britain coming closer to the Franco-German alliance, although issues such as Turkey’s EU membership bid and the future of the UK’s budget rebate remain big areas of contention.

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