Anchored by Jake Tapper, The Lead airs at 4 p.m. ET on CNN.
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By CNN chief Washington correspondent Jake Tapper
Jerusalem (CNN) – Today in Gaza, there was a show of mourning and defiance by members of the group Palestinian Islamic Jihad, which the U.S. and Israeli governments categorize as terrorist.
While Islamic Jihad is part of the group participating in cease-fire talks in Cairo, Egypt, some of its members – captured on video by a CNN crew – represent a political reality that will fight any attempts to quell violence in the region.
There are also Israelis that may be part of that effort on the other side. A right-wing faction chanted in a celebratory way about the death of Palestinian children, killed by Israeli airstrikes in Gaza, during a counter-demonstration to an anti-war rally in Tel-Aviv.
"Oleh, oleh, there is no school tomorrow; there are no children left in Gaza," they chanted.
Their actions may be quite different, but as a political force, these are the people who do not want to give peace a chance.
The Western world is more than aware of the terrorist acts of Hamas and Islamic Jihad, and their enablers throughout the Arab world.
But to many liberal Israelis, the fervor growing inside their own country is also hurting long-term prospects for peace.
"I think the extremism is here to stay. That's my main concern, because the war is almost over, but the scars will stay here," Gideon Levy, a liberal columnist with the Israeli newspaper Haaretz, told CNN.
"All this will be from now on much harder, if not complicated and dangerous," he said.
"It's horrifying to see the level of indifference in Israel toward the sacrifice, and the suffer, and the destruction, and the killing on the other side. It's almost not legitimate to even express some kind of empathy towards it," said Levy.
Prime Minister Benjamin Netanyahu's spokesman denies this is a widespread problem.
"The extremists, those making the noise, those who talk about violence and instigating sort of hatred against the other sides, they are a small minority," spokesman Mark Regev told CNN's "The Lead with Jake Tapper" in July.
But whatever their numbers, the Israeli right is flexing its muscles. As the Israeli army attacked Gaza, the strongest pressure Netanyahu may have felt was from those who thought he was holding back.
Netanyahu's deputy minister of defense Danny Danon criticized him for saying he would agree to a cease-fire. Netanyahu fired Danon, but when CNN met up with him today at an Israeli settlement south of Jerusalem, Danon was unrepentant.
"I think it was a premature cease-fire.We have to complete the operation, and uproot Hamas, otherwise Hamas will decide when the next conflict will take place. It will be next week or next month," Danon said.
"We started a war now we need to finish it," he said.
Leaders of the Israeli military cautioned the Israeli cabinet that trying to do more in Gaza would take months, cost thousands of Palestinian lives, and hundreds of Israeli lives.
"I don't buy it," Danon said. "I heard other views from the military that you can control Gaza in a week."
Danon also does not buy the argument that his obstinance and refusal to compromise ultimately hurts Israel, an argument that Israel's Western allies have tried to make.
"With all due respect, we live here. And in two and a half years Secretary Kerry will not be in the State Department. President Obama will not be in the White House. I will stay here with my family," said Danon.
Danon is a man of words, a politician. But liberals like Levy say the rising strain of extremism here is being expressed in more aggressive ways.
Recently Levy described the elite Israeli Air Force pilots of "perpetrating the worst, the cruelest, the most despicable deeds." He was defending his views on live television when a passerby accosted him – "You're a traitor!" the man shouted.
An Israeli lawmaker agreed, calling for Levy to be tried for treason.
"I was very surprised by the intensity, the level of aggression, lack of tolerance, and lack of understanding! What does it mean to live in a Democratic society?" said Levy. "My newspaper had to hire for me a bodyguard."
The question now is whether those voices – loud, angry, and sometimes violent – will derail the hopes for a long-term peace.
CNN's Katie Hinman, Eric Marrapodi, and Kim Berryman contributed to this report.