Some of the more restrained analysis, though, reveals unexpected ways to think about contemporary terrorism. Mark S. Hamm and Ramón Spaaij break down terrorists into the ways they form. In their forthcoming book, “The Age of Lone Wolf Terrorism,” they look at the solo attacker, like Jared Loughner (Tucson, 2011) and Omar Mateen (Orlando, 2016). Other studies might examine, say, cells of three or more, or pairs like the Millers in Las Vegas and the Tsarnaev brothers in Boston.

Curiously, the trigger for lone wolves, Spaaij told me, was not only political but often personal. Not long before his cop-killing spree outside Des Moines, Greene had a fight with his mother, resulting in her getting a protective order against him — a break eerily similar to that of Richard Poplawski, the Pennsylvania man whose mother called 911 on him in 2009, leading to the murder of three Pittsburgh police officers.

What often precedes these personal crises leading to violence, Spaaij said, is the “enabler” phase of radicalization, which can involve the killers’ happening upon encouraging messages from faraway leaders. When Muslims are identified as the attackers, we often learn about their obsession with the droning sermons of online imams like Anwar al-Awlaki. In the radicalization of Poplawski, “research shows that our discourse provides the environment that can enable terrorism,” Spaaij said. “He had embraced this conspiracy theory that Obama would take away his right to bear arms.” Spaaij cited conspiracy theorists like Alex Jones, who promotes the idea that FEMA camps are being built as concentration camps for dissidents and the federal government is hatching plans to seize people’s guns, as spreading the kind of degraded public discourse that lays the groundwork for action. Then, when violence occurs, “they can distance themselves and say that you cannot connect that with us because we don’t advocate violence.”

In 2010, Sarah Palin posted a map to her Facebook page that laid the cross hairs of a gun sight over districts that threatened the Republican agenda, including Gabrielle Giffords’s. Months later, when a gunman did fire a bullet point-blank through Giffords’s head, some conservative activists were outraged that anyone drew a connection between Palin’s playful clip art and an actual head shot. Donald Trump had, on at least one occasion, suggested vaguely that “Second Amendment people” knew what to do with his opponent. During the Republican National Convention, Trump’s campaign issued a statement that it did not “agree” with the view of Al Baldasaro, his delegate and a New Hampshire state representative, that “Hillary Clinton should be put in the firing line and shot for treason.” This language is routinely defended as metaphorical when spoken by prominent people, and just salty heartland gibberish when uttered by anybody else. And always, in the end, there comes the disclaimer that these are just expressions, never intended to encourage anyone to violence.

Since the election, there has been an explosion of hate speech — Trump’s name with the T bent into a swastika showed up on walls, and one racist group with the reassuringly meaningless name National Policy Institute erupted into straight-arm Nazi salutes. YouTube videos increasingly feature emboldened racists ranting in any public place — an airplane, a craft store, a coffee shop, a classroom, a school bus.

The study of terrorists’ history suggests there is a relationship between these two forms of radical discourse. Distant authorities talking in a deniably cryptic way contribute to the rationalization for violence. This shift toward violence can have an effect at just about every level — from the lone-wolf killer, to couples, to hidden cells. And although it’s a more esoteric field of study for policy-center professionals, radicalization of an entire nation is possible, too — typically after reckless innuendo from political leaders becomes acceptable and then routine.