Mr. Cotton was the one who persuaded the White House to consider Mr. McMaster, who became known over the years for questioning orthodox views of the Vietnam and Iraq wars. “Donald Trump is an unconventional president, and I think it fits him well to have someone who for many years colored outside the lines and so many times was proven right,” Mr. Cotton said.
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While some critics worry about the prevalence of military officers in political posts, others have welcomed the three generals, hoping that they will serve as a brake on bad ideas. “All three of them showed an independent streak and a talent that is in my view extraordinary, and I’ve known a hell of a lot of them over the years,” Senator John McCain, Republican of Arizona, said in an interview.
How much any of the three will be able to shape Mr. Trump’s policies remains an open question. When the president considered reinstituting torture for the interrogation of terrorism suspects, Mr. Mattis objected and Mr. Trump backed down, saying he would defer to his defense secretary. But when the White House enacted a temporary ban on refugees and on any visitors from seven predominantly Muslim countries, Mr. Kelly was not fully briefed until late in the process.
Mr. McMaster, 54, will have an office across the West Wing from Mr. Trump, and will see him more than any of the three. He has the least Washington experience of the group, meaning he will have to learn on the job how to balance the various constituencies, including the Pentagon, the State Department, the C.I.A. and Congress. He will also have to get to know a president who, Mr. Cotton said, had never heard of Mr. McMaster just a week ago.
Mr. McMaster will also have to figure out how to handle Stephen K. Bannon, the president’s chief strategist, who was given a seat on the cabinet-level national security principals committee and has played a strong role in foreign policy so far. Sean Spicer, the White House press secretary, said on Tuesday that Mr. McMaster would have authority to organize the national security team as he sees fit.
But the three generals are not known only for their experiences during war. Mr. McMaster and Mr. Mattis are both thought of as scholar-soldiers. The author of a dissertation examining the failures of the military leadership during Vietnam, Mr. McMaster has lately been running a command charged with rethinking the Army of the future. Mr. Mattis, 66, a student of history, is “just as likely to quote Cicero to you as Clausewitz,” as former Defense Secretary Robert M. Gates put it.
As the head of Southern Command, Mr. Kelly, 66, spent a lot of time thinking about how to protect the southern border and fight drug trafficking beyond traditional combat operations. He also represented the Marines on Capitol Hill, learning the byways of Congress.
“The wars clearly played a role in shaping all three men, and also certainly in shaping their reputations,” said Mr. Gates, who served Presidents George W. Bush and Barack Obama. But, he added, “These are much broader, deeper people than just their experience in Iraq, and I find that all reassuring.”
The three generals did not all overlap at the same time and place in Iraq. Mr. Mattis and Mr. Kelly grew up together in the Marine Corps, rising to four-star generals in a service that has only a handful of officers at that rank. Mr. Kelly served as assistant commander of the First Marine Division under Mr. Mattis during the initial invasion of Iraq in 2003. Mr. Kelly returned there in 2004 and a third time in 2008, when he was named the top American commander in western Iraq.
While the Marines focused on Anbar Province, Mr. McMaster and the Army’s Third Armored Cavalry Regiment concentrated on Tal Afar to the north, where in 2005 he introduced a new counterinsurgency strategy that bucked the military leadership’s thinking and helped recapture the city. His success was later cited as a model in a counterinsurgency manual that Mr. Mattis had a role in drafting with Gen. David H. Petraeus and that inspired the larger strategy shift that turned the war around in 2007 and 2008.
Mr. McMaster’s approach was not always appreciated by the military brass, however, and he was twice passed over for promotion to brigadier general. Mr. Gates personally intervened by summoning General Petraeus to take over the promotion board and ensure that Mr. McMaster would receive his first star.
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Mr. Mattis was not always in concert with superiors, either. He led the first Marines into Afghanistan after Sept. 11, but when he sought to pursue Al Qaeda into the Tora Bora region, he was not given permission, and Osama bin Laden ultimately escaped. In Iraq, where Mr. Mattis’s radio call sign was Chaos, reflecting the havoc he sought to rain down on enemies, Mr. Mattis objected when told to abort an offensive to retake Falluja early in 2004 for what he considered political reasons.
Mr. Kelly is also known for speaking his mind. While leading Southern Command, he talked about the need to rebuild aging facilities at the Guantánamo Bay prison, despite the Obama administration’s official policy that the detention center was about to close. He also called for more Navy ships to conduct counternarcotics patrols despite being repeatedly told to back down.
“He does what he thinks is right, and is anything but politically correct,” said James G. Stavridis, a retired admiral who served as NATO’s top military commander and has known the general since 1979.
If all three share common experiences from the wars of the last 16 years, they were most personally felt by Mr. Kelly. In 2010, his son, Lt. Robert Michael Kelly, was killed when he stepped on a land mine in Afghanistan, making the father the highest-ranking officer to lose a son or daughter in either war.
“They’ll bring a very sober, realistic, practical experience base to the discussions in the Situation Room because they’ve lived it,” said Douglas E. Lute, a retired three-star Army general who was a senior national security aide to Mr. Bush and Mr. Obama.
David W. Barno, another retired lieutenant general and a former commander of American forces in Afghanistan, said the three would take a more thoughtful approach to the use of force. “They’ll bring in some hard-nosed realism of what war looks like,” Mr. Barno said. “It doesn’t mean they shy away from using the military, but they will have a perspective no one else in the White House will have.”