A military power grab this week pushed Egypt into the most serious political crisis since Hosni Mubarak´s fall. As the Muslim Brotherhood was mobilizing the masses back into Cairo´s Tahrir Square, the ruling generals on Friday proclaimed their willingness to use "an iron first" to restore order.

Perhaps more than anyone outside the military, Khairat Al Shater will shape the outcome of this showdown. The millionaire businessman is the boss, in a Chicago machine sort of way, of the Muslim Brotherhood. At Wednesday´s meeting of the Islamist group´s governing Guidance Council, Mr. Al Shater sits in the middle of a packed conference table at their new headquarters in the Mokattam Hills neighborhood above the turbulent streets of Cairo.

He confesses to fatigue from lack of sleep "in these tough days"—and then exhibits little evident sign of it. At age 62, Mr. Al Shater is a tall, hulking man with a deep voice. He avoids the standard Muslim Brother costume of suit and tie and wears an open checkered shirt and blue blazer. He rarely appears in Egyptian media, but his authority goes unquestioned. If the Brotherhood came to power, Mr. Al Shater would be in charge.

The granddaddy of the Mideast´s political Islam movements has never come so close to realizing this dream. Mr. Al Shater spells out his "red lines," the minimum that the Brotherhood will accept from the military: The Supreme Council of the Armed Forces must restore the dissolved parliament, which the Brotherhood dominated. The military needs to rescind Sunday´s decree giving the generals nearly unlimited power, including over the writing of a new constitution. It needs to hand over the reins to the new president elected "freely and fairly" last weekend. The Brotherhood´s Mohammed Morsi won more votes, according to aggregate results from 13,000-odd polling stations, than Mubarak-era politician Ahmed Shafik. The regime has delayed announcing an official winner.

Listing these demands, Mr. Al Shater takes a harsher tone toward the generals than ever before. The military, he suggests, betrayed them.

"We came to think positively of the SCAF [the ruling military council] and accept its promises to share power," he says in Arabic through a translator. "Those promises were not fulfilled." After a failed attempt to let the Brotherhood form a government this spring and most of all after this week, he adds, the Brotherhood was "forced to change our position because the people of the SCAF were not intending to hand over real power."

The clash with the generals is a risky move for a legendarily cautious man. How far the Brotherhood is willing to push the military will be clearer in coming days. Coming together again with the young activists, it flexed its muscle by mobilizing 200,000 or so into Tahrir Square on Tuesday at a few hours´ notice. Friday´s protests were as big.

Yet prudence elbows in on Mr. Al Shater´s uncharacteristic belligerence. The Brotherhood isn´t mobilizing trade unions. While crowds in Tahrir are a nuisance, national strikes could paralyze a weak economy and harm business elites, including the military. Mr. Al Shater repeatedly invokes variations of "we don´t want a collision" with the regime—be it to reform the security services, overhaul the command-style economy or resolve this conflict. Egypt can only be changed "gradually," he says. "It will take two, three, four, five years." Put this way, his red lines get blurrier.

"Politics is the art of the possible," he says. "Don´t expect us to look for a collision or use violence. Our method is the peaceful way, relying on the basics of the political game. We tried dialogue and we will try dialogue now and in the future."

This hasn´t always been true. Founded in 1928 as an Islamic social and political movement, the Brotherhood was repressed in the late 1940s, more brutally in 1954 two years after the Nasserite military coup, and throughout the Mubarak era. Some of its former members have turned to violence, including al Qaeda´s current leader, Ayman Al Zawahiri, who was among those who plotted the assassination of Mr. Mubarak´s predecessor Anwar Sadat in 1981.

In the late 1990s, Mr. Al Shater negotiated a deal to let the group run for a select number of seats in parliament. When the Mubarak government changed its mind about allowing the Brotherhood into politics, Mr. Al Shater in 2007 ended up back in prison, his fourth stint. Released a month after President Mubarak´s ouster in February 2011, he declared that the Brotherhood´s political ambition was "participation, not domination" in a freer Egypt´s politics.

The group has since tried to strike a balance between pushing for power and avoiding a return trip behind bars. It has played both sides, and annoyed both too. The secular democracy activists who reluctantly joined this week´s protests worry that the Brotherhood will, just as it did after the fall of President Mubarak, reach its own accommodation with the military regime, at the expense of real democratic reform. Younger and ambitious Brotherhood members who challenged Mr. Al Shater´s dominance within the organization were kicked out and started new moderate Islamist parties. The secular establishment in business, the large state machinery, the security services and courts are as anxious as ever about an Islamist ascendancy. And in an Egypt whose cities are full of vibrant coffee shops and bars open all night, the Brotherhood´s secrecy and religious orthodoxy set it apart from the mainstream.

Looking back on the past 16 months, Mr. Al Shater says he regrets "disagreements over methods" with other anti-government groups. The Brotherhood has toned down its religious rhetoric. As part of a new outreach effort, Mr. Al Shater calls for a "broad-based coalition" to put together "a national accord" to bring democracy and rule of law to Egypt.

The Islamists, who were the only organized opposition force to survive the Mubarak era, last year pledged to abjure a parliamentary majority and not field a presidential candidate. The group broke both promises. Mr. Al Shater says the military´s refusal to share power forced them to change tack. When the regime disqualified him as a candidate for president, the uncharismatic Mr. Morsi stepped in.

The decades of repression pushed the Brotherhood underground, but the group has started to open up in the changing era. A new legal political party was founded. Luxury cars are parked outside the new headquarters and a large green banner in Arabic and English hangs from the side with the name and symbol of the Brotherhood.

Yet the Brotherhood remains a closed, rigidly hierarchical and disciplined quasi-Trotskyite organization. Still not legal, the group has yet to reveal its membership rolls or finances. According to a former leader, it has over half a million members—about 75,000 of whom are educated professionals. Doctors, engineers and private- sector businessmen, rather than religious scholars, control the group.

Formally the deputy general guide, Mr. Al Shater heads the dominant conservative wing of the Brotherhood—also known as the "Persian Gulf" crowd. The son of a merchant, he dabbled in Nasserite Socialist student politics and gravitated to Islamism in the 1970s during Sadat´s rule. After Sadat´s murder, he joined many Islamists who went into exile in the Gulf—in his case to Saudi Arabia. He returned a devout Muslim and Brotherhood leader. His wife wears a niqab covering her face, which is atypical in Egypt. A fortune made in textiles, furniture and software allowed him to become the Brotherhood´s chief financier. His intelligence and presence are the other sources of his political authority.

Egyptian jokes refer to Mr. Morsi as "a spare tire," or worse. By broad consensus, a President Morsi would be subservient to Mr. Al Shater and a Brotherhood that is still technically illegal. When I make this point, Mr. Al Shater waves it off with three "nos" but little other explanation. He slaps at a fly with one of his large hands. "The Muslim Brotherhood is one of the most democratic organizations in Egypt," he adds, saying decisions are taken by a consultative "shura," or council.

By tactical choice or shifting priorities, the Brotherhood keeps its political focus away from divisive social or religious issues. "The economics file is the most important," says Mr. Al Shater. Detailed policy papers and a shadow cabinet were put together to prepare the Brotherhood to govern a country of 81 million. The group´s economic views place them on the free-market end—or as Mr. Al Shater says, "Our Islamic vision is close to a modified capitalism . . . to mix an open market and take care of social welfare."

The nod to charity reflects the Brotherhood´s traditional emphasis on social work, which explains its popularity among poorer Egyptians. The bigger priority is to break up, gradually" in his words, the old state patronage networks and give private business room to thrive. A richer Egypt will be a stronger country, he says, mentioning Turkey and Malaysia: Muslim countries with booming, open economies. Yet he doesn´t rule out the possibility that the ultrareligious Salafists might get education or social-policy portfolios, which is one of the fears of secular Egyptians and rights activists.

If handed executive power, will the Brotherhood try to Islamicize Egypt from the top down? "Who says this doesn´t know the nature of Islam," shoots back Mr. Al Shater. "There´s a phrase in Islam: ´You should not force anything in religion.´" These issues can be handled "in a very democratic way" and safeguard free choices "in personal life." So it would seem that under a Brotherhood government, beer or bikinis may be banned by an elected parliament but enjoyed in the privacy of an Egyptian´s home.

The Camp David peace treaty falls into a similar gray area. Mr. Al Shater says the Brotherhood never said it would call a referendum on the treaty, without explicitly ruling it out. "Egypt is a constitutional country," he says. "It´s not possible with each ruler to rethink our obligations and conventions. This will lead to a lack of confidence in dealing with Egypt." The Brotherhood has a close alliance with Hamas, and its views on Israel are no secret. But he says the priority is a close "strategic partnership" with the U.S., which the group expects to help it unlock credit markets and gain international legitimacy.

How the Muslim Brotherhood might rule is a hypothetical as long as the Egyptian military holds the levers of power. Sure of its presidential win and willing to call protesters into the streets, the Brotherhood may not see a better opportunity to run the world´s largest Arab state than the current moment.

Mr. Al Shater is a careful strategist who knows his adversaries in uniform intimately. He is also not a revolutionary who will take risks gladly. The real action may not be on Tahrir Square, but in his offstage dance with the generals.

Mr. Kaminski is a member of the Journal´s editorial board. (Copyright © Dow Jones & Company, Inc.) 06/23/12)