RUTHFULLY YOURS

Identical in style and details to earlier execution videos released by the Islamic State, this one was the first the group has released depicting a killing outside of its core territory in Syria and Iraq. It appeared to show much closer communication and collaboration between the Islamic State, also known as ISIS or ISIL, and its far-flung satellite groups than Western officials previously believed.

As the Obama administration seeks broad approval to use military force in an open-ended war against the Islamic State, the new video may reinforce the concerns among some lawmakers that the legislation could authorize operations in unexpected territories like Libya, where local militants are planting the Islamic State flag as “provinces” of the group.

Concern is already growing in Libya and the West that the group might capitalize on the chaos that has engulfed the country in order to establish and expand a base of operations there. At least three groups of Libyan fighters have already pledged loyalty to the Islamic State, one in each of the country’s three regions: Barqa in the east, Fezzan in the south and Tripolitania in the west.

Officials of Libya’s internationally recognized government recently traveled to Washington to seek help from the West in preventing the Islamic State’s expansion. Even some opponents fighting that government as part of a coalition with Libyan Islamist factions have reportedly begun raising alarms about the need to stop the Islamic State from expanding in Libya.

In Cairo, where the military-backed government has been working to defeat the Islamist factions in neighboring Libya, supporters of the government cited the video released Sunday as new evidence that those factions pose a growing threat to Egypt’s own security.

Confirming that those killed in the video were Egyptian Christians taken hostage in Libya weeks ago, President Abdel Fattah el-Sisi of Egypt on Sunday announced seven days of national mourning and a meeting of his defense council. In a televised address, he said Egypt would choose the “necessary means and timing to avenge the criminal killings.”

The White House offered support to Egypt’s government and condolences to the victims’ families in a statement on Sunday night, condemning the “despicable and cowardly” killings and saying, “ISIL’s barbarity knows no bounds.”

The Islamic State promoted the video last week with a photograph from the scene that appeared in its English-language online magazine, Dabiq.

The main difference from other execution videos it has released is that the new one appears to have taken place on the southern shore of the Mediterranean, on a rocky beach said to be in western Libya, far closer to Europe than sites previously depicted.

Fighters under the banner of the Tripolitania Province of the Islamic State announced last month that they were holding about 20 Egyptian Christians, or Copts. A similar number of Egyptian Christians in Libya seeking work had disappeared in the mid-coastal city of Surt. Officials of both the Egyptian government and the Coptic Church confirmed that captives seen in a photograph with the announcement were the missing Egyptian Christians, and on Sunday confirmed that they were killed in the video.

In the video, masked fighters identified as from the Tripolitania Province of the Islamic State, dressed in black with machetes at their chests, parade along a rocky beach toward the camera with a row of bound captives in orange jumpsuits, like the ones worn by victims in previous Islamic State videos.

About five minutes long, the video bears the logo of Al Hayat, the Islamic State’s media arm. Unlike the cellphone videos usually made by Libyan militants, it is as polished as previous Islamic State videos, with slow motion, aerial footage and the quick cuts of a music video. The only sound in much of the background is the lapping of waves.

The captives are made to kneel in the sand. Then they are simultaneously beheaded with the theatrical brutality that has become the trademark of Islamic State extremists. There was no indication in the video about when the beheadings took place.

The lead executioner speaks in fluent English with an American accent, and his words are translated in Arabic subtitles. Under the title “A Message Signed With Blood to the Nation of the Cross,” he emphasizes that the fighters are just one part of the broader Islamic State group.

“Oh, people, recently you have seen us on the hills of as-Sham and Dabiq’s plain, chopping off the heads that have been carrying the cross for a long time,” he said, using Arabic terms for places in and around Syria. “Today, we are on the south of Rome, on the land of Islam, Libya, sending another message.”

He implies that they are taking revenge for the killing of Osama bin Laden by American commandos and his burial at sea, saying, “The sea you’ve hidden Sheikh Osama bin Laden’s body in, we swear to Allah we will mix it with your blood.”

One subtitle adds that the killing is also retaliation for a sectarian dispute that flared in Egypt five years ago over a Coptic Christian woman, Camilia Shehata, the wife of a Coptic Christian priest. She disappeared for a time, and many Muslims believe she tried to convert to Islam, only to be kidnapped by her husband and members of the church.

Ms. Shehata briefly became a cause célèbre among Islamist militants, before the Arab Spring eclipsed such skirmishes.

“This filthy blood is just some of what awaits you, in revenge for Camelia and her sisters,” a caption declares, as blood from the prisoners darkens the waves.

Analysts said the video challenged the presumptions of many Western analysts that militants in places like Libya might be adopting the banner of the Islamic State for its notoriety without signing on to its bloodthirsty and messianic ideology.

“It is one thing to fly the ISIS flag because a lot of guys are doing it,” said William McCants, a researcher at the Brookings Institution who studies Islamist militants. “It is another thing to capture a bunch of Egyptian Copts and kill them and see it as some of part of a grand, final-days battle.”