A parallel can be seen with the American refusal to speak to the Palestinian party Hamas, which decisively won elections in the West Bank and Gaza in 2006. Asked what role Hamas would have in the renewed talks, Mr. Mitchell answered with one word: “None.” No serious analyst believes that peace can be made between Palestinians and Israelis without Hamas on board, any more than could have been the case in Northern Ireland without Sinn Fein and the I.R.A.

The United States insists that Hamas meet strict preconditions before it can take part in negotiations: recognize Israel, renounce violence and abide by agreements previously signed between Israel and the Palestine Liberation Organization, of which Hamas is not a member. These demands are unworkable. Why should Hamas or any Palestinian accept Israel’s political demands, like recognition, when Israel refuses to recognize basic Palestinian demands like the right of return for refugees?

As for violence, Hamas has inflicted a fraction of the harm on Israeli civilians that Israel inflicts on Palestinian civilians. If violence disqualifies Hamas, surely much greater violence should disqualify the Israelis?

It was only by breaking with one-sided demands that Mr. Mitchell was able to help bring peace to Northern Ireland. In 1994, for instance, Mr. Mitchell, then a Democratic senator from Maine, urged President Bill Clinton  against strenuous British objections  to grant a United States visa to Gerry Adams, the Sinn Fein leader. Mr. Mitchell later wrote that he believed the visa would enable Mr. Adams “to persuade the I.R.A. to declare a cease-fire, and permit Sinn Fein to enter into inclusive political negotiations.” As mediator, Mr. Mitchell insisted that a cease-fire apply to all parties equally, not just to the I.R.A.

Both the Irish and Middle Eastern conflicts figure prominently in American domestic politics  yet both have played out in very different ways. The United States allowed the Irish-American lobby to help steer policy toward the weaker side: the Irish government in Dublin and Sinn Fein and other nationalist parties in the north. At times, the United States put intense pressure on the British government, leveling the field so that negotiations could result in an agreement with broad support. By contrast, the American government let the Israel lobby shift the balance of United States support toward the stronger of the two parties: Israel.