The prisoners exchange that Hamas and Israel concluded this week could be a potential historic turning point in an otherwise moribund “peace process” where no noteworthy breakthrough has occurred in the past nearly 20 years of American-mediated, and therefore mostly Israeli-defined, talks.

The prisoners exchange is significant for showing that the most implacable and violent enemies are able to negotiate and reach agreement, when both sides obtain gains that are sufficiently important for them to be able then to make concessions on issues of equal importance to the other side.

This deal builds on previous Hamas-Israel cease-fire agreements. It shows that leaders on both sides can actually lead their people, go against skeptical public opinion, and consummate a deal. Finally, it shows that both leaderships win serious new public acclaim and political strength from their agreement, which only strengthens their legitimacy, paving the way – potentially – for other agreements that can achieve meaningful gains for both sides.

After the exchange, mass celebrations took place in Gaza and the West Bank, and 79 percent of Israelis approved the agreement (according to Yedioth Ahronot). Israeli and Palestinian politicians who savor this moment should seriously consider ways to build on their breakthrough for further mutually meaningful gains.

They should do so because of the enormous contrast between this breakthrough and the genuine popular acclaim, even jubilation, it generated on both sides, and the hopeless approach of the American-Israeli-directed Quartet’s attempt to resume direct negotiations between Benjamin Netanyahu and Mahmoud Abbas. Even the Palestinian prime minister, Salam Fayyad, said in Washington this week that the Quartet attempts to push for parallel talks with both sides is “meaningless,” given the newly announced Jerusalem settlement of Givat Hamatos and the helplessness of the Palestinians to freeze such colonization.

If Israel is able to negotiate indirectly with Hamas and reach a deal that pleases both sides, why does the Quartet continue to succumb to the American refusal to engage Hamas diplomatically? This is all the more bizarre in view of the fact that even some Israelis voices are calling for a policy review. The senior Kadima politician Nachman Shai a few days ago called for a lifting of the Gaza blockade and a rethinking of the economic model in Gaza. Shai urged Israeli leaders to take advantage of the prisoners agreement, and rethink Israel’s stance on Hamas and the blockade, suggesting that, “With Gilad Shalit’s return home, Israel needs to weigh the possibility that relations with Hamas may be open to change,” and Israel had to, “re-evaluate its stance toward Hamas as part of a renewed attempt to restart peace negotiations with the Palestinian Authority.”

The U.N. agency responsible for Palestinian refugee basic services, UNRWA, which knows the reality on the ground in Gaza and Israel better than any other third party, similarly called for a policy reassessment. UNRWA spokesman Chris Gunness said the prisoners deal offered, “an ideal opportunity for Israel to conduct a thorough review of the policy of blockading Gaza. It is surely time to think again about policies that have driven the Gaza economy underground and into tunnels. We must all work to create an economic model in Gaza that is based on legality, full employment, prosperity, stability and peace. That is surely what most Israelis and people in Gaza want. A prosperous Gaza, like a prosperous West Bank, is in the interests of all.”

Rethinking the old policy, removing the Gaza blockade, and easing Israeli-Hamas violence and political tensions would also benefit Egypt, a crucial peace partner and security ally for Israel. The prisoners exchange may be the opportunity that many have been seeking to allow the Quartet to shift from flogging a dead horse to nurturing a new diplomatic seed that seems ready to sprout. It is possible to imagine a deal that includes a new cease-fire agreement, mutual acknowledgment of each other’s “legitimate security concerns,” ending the Israeli blockade of Gaza, and imposing a new border regime by Egypt which “regulates” the Rafah crossing, deals with the smuggling tunnels, and also tightens issues of security in the Sinai.

This is a long shot, and full of obstacles. It would be hard to pull off – but not any harder than the prisoners swap that Israel and Hamas just pulled off successfully, to wide public acclaim.

With such a daring new approach, the Quartet could rehabilitate itself by acting more seriously, Israeli and Hamas leaders would both gain significantly, their publics would both welcome achieving their core aims through peaceful negotiations, a new momentum would push Hamas and Fatah into a more serious discussion of reconciliation and Palestinian unity, Egypt would continue to resume its dignified and productive role in the diplomatic world, and the United States would discover a plausible way to climb down from its failed policy of boycotting Hamas and instead enjoy the fruits of actually acting with wisdom and realism in the Middle East.

It is rare that the Arab-Israeli conflict offers all sides such a win-win opportunity. May God and Elvis offer all concerned the wisdom to recognize this and act accordingly.