The Supreme Court on Thursday heard historic arguments on whether former President Donald Trump can be criminally prosecuted related to his efforts to overturn his 2020 election loss.

He denies all wrongdoing and insists he should have "absolute immunity" for any "official acts" while in office.

His name and the specifics of the case were rarely mentioned -- just a handful of times -- as the justices, over the course of nearly three hours, grappled with questions about what their ruling will mean for the future of the presidency and how that should be reconciled with the principle of the rule of law.

The former president, who lamented not attending arguments in Washington as he is on trial in New York in a separate criminal trial, in which he has pleaded not guilty, was represented by attorney John Sauer.

Michael Dreeben argued on behalf of the government and special counsel Jack Smith. Smith last year brought four felony counts against Trump, including conspiracy to defraud the U.S. and obstruction of an official proceeding, for his attempts to remain in power after losing to now-President Joe Biden.

Here are five key takeaways from the arguments.

Hypotheticals like coup, assassinations dominate

The question of presidential immunity sparked an array of "what if" scenarios from the justices as well as multiple citations to past presidential choices ranging from what was done by Franklin D. Roosevelt, Richard Nixon, George W. Bush, Barack Obama and others.

Justice Sonia Sotomayor, minutes into the arguments, raised the question of whether criminal immunity would extend to a president ordering the military, or someone else, to assassinate a political rival if they believed this person was "corrupt"

Sauer affirmed that such a decision could "well be an official act" excluded from prosecution, telling Sotomayor, "It would depend on the hypothetical."

At another point, Justice Elena Kagan asked the Trump attorney: "How about if a president orders the military to stage a coup?"

Sauer said that, likewise, under her hypothetical, that could "well be" official behavior of the president though "it would depend on the circumstances."

"It certainly sounds very bad," he conceded, "and that's why the framers [of the Constitution] have -- and that's why the framers have a whole series of structural checks that have successfully, for the last 234 years, prevented that very kind of extreme hypothetical, and that is the wisdom of the framers."

He maintained that the only time a president could be criminally prosecuted was if they were impeached by the House and convicted by the Senate for the same crime.

Even then, Sauer later suggested, there would have to be a criminal statute against coups expressly referencing the president.

Some justices also raised questions about whether presidents could pardon themselves, and both attorneys said it's an issue that's never been addressed before and has no precedent.

Private vs. official acts and concessions from Trump's attorney

Central to Thursday's arguments was what would qualify as an "official" act of the presidency that may be protected by a finding of immunity versus what constitutes private conduct.

In one instance, Justice Amy Coney Barrett asked a series of probing questions of Sauer about conduct alleged in Smith's indictment and whether Trump's attorneys considered them private or official acts.

Sauer conceded some acts in the indictment were unprotected, including his allegations that Trump turned to a private attorney who knowingly spread false claims of election fraud to spearhead his challenges to 2020 election results and that he conspired with others to implement a plan to obstruct the certification of President Joe Biden's win.

Justice Kagan pressed Dreeben about what the government would consider official and unofficial, or private, conduct. Dreeben said core executive functions, like pardon power, vetoing legislation, foreign recognition and more have absolute protection.

But organizing a false slate of electors, a key allegation in Smith's indictment, is "campaign conduct" and not official.

"That's not any part of a president's job," Dreeben said.

Conservatives worried about bad faith prosecutions

Several conservative-leaning justices expressed concerns about the potential for bad faith prosecutions against a president.

Chief Justice John Roberts raised concern about the lower court ruling, specifically that its position could put too much faith in the justice system to act non-politically and out of good faith.

DOJ's Dreeben responded that there are "layered safeguards" that protect against malicious prosecution.

"We are not endorsing a regime that we think would expose former presidents to criminal prosecution in bad faith, for political animus, without adequate evidence, or politically driven prosecution that would violate the Constitution," Dreeben said.

Justice Brett Kavanaugh suggested concerns about a vicious cycle of malicious prosecutions hampering presidents for years to come. He also raised the question of the "risk" of a "creative prosecutor" using "vague" statutes -- including obstruction and conspiracy, which Trump faces -- against a commander in chief.

Justice Samuel Alito also appeared skeptical of Smith's use of at least some of the conspiracy and fraud-related charges, asking if the conspiracy statute wasn't "peculiarly open-ended."

In response, Dreeben said: "It is designed to protect the functions of the United States government, and it is difficult to think of a more critical function than the certification of who won the election."

Will the court send the case back?

Several conservative justices also raised the question of whether to remand the case back down to the U.S. Court of Appeals for the District Court of Columbia (which rejected the immunity claim in sweeping terms) to conduct further proceedings.

Remanding the case would -- perhaps significantly -- delay Trump's Jan. 6 trial even past the November election. The trial was set for March 4 but is on pause until the immunity question is resolved in the judicial system.

Justice Neil Gorsuch was the first to suggest further proceedings could be required to determine what is an official act and what is private and thus more open to prosecution.

"What concerns me is, as you know, the court of appeals did not get into a focused consideration of what acts we're talking about or what documents we're talking about," Chief Justice John Roberts said.

Sotomayor pressed back on the notion of remanding the case, arguing that even in the instances of acts that could be considered official, they came in the context of Trump pushing forward in his "private" intent of remaining in office despite losing the election.

'A rule for the ages'

Justices emphasized throughout arguments that they were not only considering the immediate case involving Trump, but what these questions mean for the future of the country.

"We're writing a rule for the ages," Gorsuch said.

Justice Samuel Alito noted whatever that the high court decides "is going to apply to all future presidents."

The criminal immunity question is a novel one, as Trump is the first president (current or former) to be criminally charged.

The closest a former commander in chief came was President Richard Nixon, who resigned in 1974 rather than be impeached and was later pardoned by his successor -- which was mentioned by the justices during Thursday's hearing.

The justices grappled with that unprecedented nature and what it would mean for a president to have too much or too little protection for any actions taken while in office, including perpetually controversial decisions like those amid war or when addressing national security.

"If the potential for criminal liability is taken off the table, wouldn't there be a significant risk that future presidents would be emboldened to commit crimes with abandon while they're in office?" Justice Ketanji Brown Jackson said.

But Sauer indicated that such problems hadn't occurred so far.

"I respectfully disagree with that because the regime you described is the regime we operated under for 234 years," he said.

ABC News' Alexander Mallin and Mike Levine contributed to this report.