What was surprising about President Obama’s denunciation of Donald Trump’s anti-immigrant terrorism policy Tuesday wasn’t the fact that Obama never mentioned Trump by name. He didn’t have to. “We now have proposals from the presumptive Republican nominee for President of the United States to bar all Muslims from emigrating to America,” an angry Obama declared as the nation’s emotions remained raw from Sunday’s slaughter of 49 people in Orlando, by a killer who professed support for radical Islam. “We hear language that singles out immigrants and suggests that entire religious communities are complicit in violence. Where does this stop?”

What was surprising was that the nation’s top military officer stood by the President’s side as the commander-in-chief issued an extraordinary public rebuke of the man seeking to be that general’s next boss. Washington corridors were abuzz Wednesday after the ramrod-straight and uniformed presence of Marine General Joseph Dunford, chairman of the Joint Chiefs of Staff, provided, wittingly or not, a beribboned endorsement of Obama’s condemnation of Trump.

“If you’re going to make a political speech, and that’s what this was, you should not have the chairman, or anybody in uniform, standing next to you as a potted plant,” says Anthony Zinni, a retired four-star Marine. “It gives the appearance that he’s in agreement with the political statements that were made.”

Zinni and other former senior officers believe Dunford was uncomfortably trapped after Obama’s ISIS remarks drifted into the political realm. Dunford’s spokesman wouldn’t go that far, simply saying that the chairman “remains apolitical” and routinely appears with Obama when the President makes comments following such ISIS sessions. Tuesday’s appearance, Navy Captain Gregory Hicks said, “was no different.”

There is private concern inside the U.S. armed forces about Trump’s fitness for command, but good luck getting anyone to say so on the record. The military is built upon what it calls “good order and discipline,” a pair of attributes rarely attributed to Trump. While a recent unscientific independent Military Times newspaper survey had Trump outpolling all-but-certain Democratic nominee Hillary Clinton by a greater than 2-to-1 margin, many view that as a desire for change rather than support for Trump. “There is concern in the military about Trump, but they really want a change,” Zinni says. “There’s a lot of unhappiness about the plight of the military, the funding and the way wars are being prosecuted.” According to this thinking, Clinton is unlikely to make such wholesale changes in the U.S. military.

U.S. voters always want to know that their potential Presidents have the confidence of the 1.4 million Americans in uniform. For the Pentagon, the best way to do that is to say it respects the will of the voters, and will salute smartly no matter who is in the Oval Office. Both those in uniform and their civilian overseers stress their desire to steer clear of politics.

Defense Secretary Ash Carter, for example, stiff-armed a reporter Monday who asked about Trump’s statement that NATO is “obsolete” (like Obama, the reporter didn’t name Trump). “I’ve got to be very strict in this regard,” Carter said. “You frame a question in terms of the presidential campaign and I’m just not going to answer a question that’s framed in that way.”

General Mark Welsh, who as the Air Force chief of staff is a member of the Joint Chiefs, said Wednesday that he and his fellow chiefs don’t fret about politics and campaigns. “I want a commander-in-chief who will listen to us, I want a commander-in-chief who values the opinion of the chairman of the Joint Chiefs,” he tells TIME. “I’d like a commander-in-chief who’s thoughtful on issues and is willing to engage in discussion, and a commander-in-chief who has a team around him that feels the same way.”

Might that rule out Trump? “I’ve never met him; I don’t know,” Welsh says. “I’ll tell you, if you watch political campaigns over time, there’s a whole lot of behavior that you would kind of question, I think, not just from any one particular candidate.” The military, he adds, doesn’t play favorites. “Our job is to support the commander-in-chief as much as we can, and whoever walks into the White House is going to get that support from all of us.”

But retired officers don’t have to watch their words so carefully. “Mr. Trump is a potential disaster as commander-in-chief—uninformed, volatile, poor judgment,” says Barry McCaffrey, a retired four-star Army officer. “Hard to believe this is the candidate of a major political party.”

“Trump is unexpectedly increasing my enthusiasm for Hillary,” adds retired general Merrill McPeak, who, like Welsh, served on the Joint Chiefs as the Air Force chief of staff. “What he is saying is not based on facts: it’s based on immaturity, bad judgment and ignorance, and I think it’s going to be hard for people in uniform who are thoughtful about this, to vote for him.”

The military’s aversion to partisan politics is on display nearly every January, when the President heads to Capitol Hill to tell the Congress, the Supreme Court justices, the Joint Chiefs and the American people about the Administration’s plans for the coming year. It’s an intensely political event, with Democrats and Republicans either cheering wildly or sitting stone-faced, depending on whether or not the chief executive belongs to their party. “The service chiefs and the Supreme Court justices can’t applaud any of the lines,” Zinni says. “Fortunately, if you’re a combatant commander”—which Zinni was—“you don’t have to worry about that Washington bullshit.”

Actually, it’s a little more complicated than that. The kabuki dance begins with the chairman of the Joint Chiefs keeping a close eye on the chief justice, McPeak recalls. “The chief justice is a separate branch of government, and if he stands and applauds, then the chairman will stand up and applaud, and the chiefs will follow,” he says. “The chief justice was considered the guy who was neutral, and who knew when to stand up and applaud.”