After all the warnings and all the clichés about a war that would “spill” over Syria’s border, the savage fighters of Abu Bakr al-Baghdadi’s Sunni Muslim “caliphate” have at last arrived in Lebanon.

So far, the Lebanese army has lost 13 of its soldiers in a costly battle with rebels to retake the north-eastern Sunni town of Arsal – on the Syrian border and hitherto a resupply base for Islamists trying to overthrow President Bashar al-Assad – while the conflict has generated the same gruesome events which followed Islamist victories in Iraq and Syria: reports of civilian executions, government soldiers taken hostage, at least 12 civilians confirmed dead, including five children, and the prospect of long and bloody fighting ahead.

The world’s attention, of course, has been concentrated on the slaughter in Gaza. In the Middle East, tragedy must come one day at a time, so the Syrian civil war and the Isis takeover of western Iraq continued in the shadows of the Palestinian-Israeli conflict. But the Islamists’ arrival in Lebanon and the prospect of a mini-civil war around Arsal – and perhaps as far as Tripoli – could have repercussions far graver than the Gaza war. As Islamists take over Lake Mosul and other districts from the Kurds in northern Iraq and press harder against Syrian government troops, their extension into Lebanon marks their furthest progress yet from the Tigris towards the Mediterranean. In Arsal, the fighters – officially from el-Nusra, whose own members are already joining those of Abu Bakr al-Baghdadi’s caliphate – adopted their usual practice of seizing large buildings in the centre of the town (in this case, the technical college, a hospital and a mosque) and clinging to them in the hope that their opponents would disintegrate. The Lebanese army, which has twice defeated Islamist rebellions inside Lebanon in the past 15 years, claimed to have retaken the college, but the statements from both the Lebanese commander and Prime Minister may be taken as accurate: that the takeover of Arsal had been planned long in advance and is part of a far greater rebel strategy.

The Lebanese army says it has so far killed 50 fighters – a tally that sounds very like the Syrian army’s premature claims of victory on the other side of the border – but government forces in Lebanon are unlikely to fall back. Sunni Muslims make up the larger part of the Lebanese forces whose units are among the best integrated of Middle East armies – and this has never prevented them from attacking and subduing Sunni Muslim rebels in the past, first at Sir el-Diniyeh in the northern mountains in 2000, and then within the Palestinian camp of Nahr el-Bared in 2007, at a cost of almost 500 dead soldiers, fighters and civilians.

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For more than a year, the Lebanese army has tried vainly to close the frontier east of Arsal, and a Syrian army victory over rebels in Yabroud on the other side of the border earlier this year suggested that Sunni insurgents might leave Arsal lest they be cut off. But their resurgence shows that the Syrians have nothing like the control they have been claiming in the frontier lands. Indeed, the Nusra men had no difficulty in seizing 15 soldiers and almost as many Internal Security Force personnel when they first struck at Arsal. A battle between those Sunni forces opposing the Assad regime in Damascus – who are also responsible for the bombing of Shia targets in Lebanon – and Lebanese troops was almost inevitable. Less than two weeks ago, Lebanese special forces in Tripoli killed Mounzer el-Hassan, a Sunni “jihadist” logistics officer who was reported to have given suicide belts to bombers who attacked Beirut’s Shia southern surburbs and the Iranian embassy in the capital. Those present at the battle said that el-Hassan was playing taped Islamic music as he finally died, when a hand grenade – possibly in his own possession – blew up in his face.

His death followed shortly after the capture of Houssam Sabbagh, a Salafist militant who led Sunni militia forces in recent battles against Alwite Shias in Tripoli. Sabbagh, who has fought in Afghanistan, Chechenya and in Iraq against US forces, was one of the few Tripoli leaders who refused to participate in a government “security” plan for the city.

The battles in Syria, however, are more complex. While Isis – which still uses its acronym of the Islamic Army of Iraq and the Levant despite its incorporation into what al-Baghdadi calls the “Islamic State” or caliphate – has strengthened its position in Deir el-Zour and neighbouring villages (with its usual ferocious executions and heads-on-stakes), the Syrian military seems intent on blasting rebels out of the Damascus suburbs, especially Douma, a district which lies close to the main road north of the capital. If al-Baghdadi’s men are fighting for control in the east of the country, Assad doesn’t want them taking the place of less spirited rebels around Damascus.

Reports of independent resistance groups who oppose both Assad and Isis – and supposedly call themselves “White Shrouds” – should be taken with the usual Syrian caution. Various militia outfits of both Sunni and mixed persuasions have strode fitfully onto the stage of the civil war over the past two years, only to vanish or merge into larger rebel or government forces.

But just as they must abide by tribal rules in Iraq, Islamists have found it dangerous to take on individual Syrian tribes in the “Jazeera” plateau north of Deir ez-Zour. They may have no love for Assad, but they will not allow fighters from Algeria or Chechenya to rule their tribal lands.

More disturbing, however, is the news that Sunni gunmen from the caliphate may have taken Iraq’s largest dam outside Mosul from Kurdish peshmerga guerrillas.

The Kurds enlarged their territory by perhaps 40 per cent when the Iraqi army fled the northern Iraqi city, but the reputation of their supposedly unconquerable peshmerga army is taking a battering now that they have admitted losing villages close to the dam.