Every war, as a practical-minded American general named Eisenhower once noted, will surprise you. Cry havoc, let slip the dogs of war, and whom they will turn on is never as clear as the armchair generals, or even the real ones, may imagine.

And this Arab-Israeli war may be different from all the others. For one thing, it isn’t an Arab-Israeli war, not yet, but an Islamist-Israeli war. Surprisingly, the Arab League withheld its automatic stamp of approval for any and all assaults on Israelis.

Can it be that autocratic Arab rulers like those in Egypt, Jordan, and Saudi Arabia have realized they have more to fear from jihadist outfits like Hezbollah — and its backers — than the Israelis do?

What backers? Syria’s fingerprints, and Iran’s as well, are all over the rockets landing hourly in Nahariyah, Haifa, Nazareth . . . and all around northern Israel.

Meanwhile, across the border, caught between Hezbollah and Israel, a battered, burning Lebanon is ceasing to be a country and is becoming a battlefield. And this war may have only begun. The skirmishes along the border herald a full-scale ground war.

Things do change in the Middle East and in the crises there: This time America isn’t playing the role of Honest Broker, aka Uncle Sucker. Washington hasn’t joined the reflexive cry for stopping the Israeli offensive ASAP. In that regard, George Bush seems a different kind of president — one interested not so much in putting an end to this immediate crisis as seeing that it not recur in the future.

How? By helping to ensure that, whenever this unpleasantness ends, Hezbollah will no longer be in a position to stage a repeat performance of the attacks that triggered this one.

That’s a tall order. Just how do Washington and Jerusalem propose to achieve such a goal? There’s no secret about that: Ideally, Hezbollah’s militia would be cleared out of southern Lebanon, disarmed and replaced with another force — like the Lebanese army. Or maybe with an international contingent that, unlike the U.N. force on the border now, might actually be of use. And, oh, yes, those Israeli hostages would be returned unharmed. (Do you think they’re still alive?)

Whether all this will prove one more disaster in the history of the Middle East or a victory for peace and stability there could depend on how much time the Israelis are given to end, or at least loosen, Hezbollah’s grip on southern Lebanon. The U.N.’s role, as always in the Middle East, will not be to prevent a war but to confirm what the war has wrought on the ground.

At this point Israel’s most dangerous adversary is not Hezbollah, but time. Can the Israelis achieve their military objectives before Hezbollah is saved by the bell, — or, rather, by a cease-fire? Washington is about to seek one, but slowly.

Once again we are about to witness some shuttle diplomacy in the Middle East, and, if the American secretary of state is as smart as she doubtless is, we’re going to see more shuttling than diplomacy — at least for the next week or two, or however long it takes to push Hezbollah back from Israel’s border.

Hezbollah, however, has other ideas. Indeed, it has a whole different concept of what’s going on. Convinced it is winning, it may cooperate in its own slow attrition by stalwartly refusing to retreat, disband or hand over its hostages. Syria and Iran may be prepared to fight to the last Hezbollah fighter — a consummation devoutly to be wished.

Hezbollah may think it can go on lobbing those made-in-Iran missiles from now till doomsday — but in the Middle East doomsday is always just around the corner. For any or all the warring parties.

Hezbollah’s strategic calculus could yet prove sound if the Israelis and Americans cave. But its leader, Hassan Nasrullah, may be making a slight miscalculation, much like Gamal Nasser’s in 1967 when he blockaded the Jewish state and dared the Israelis to do something about it. Which they did.

It’s going to be interesting to see if Hezbollah’s assessment of the balance of forces is right. Interesting, but terrible. Because, as this conflict’s end game is played out around the polished table of the U.N. Security Council and diplomats with clean fingernails and muted voices negotiate the exact language of solemn resolutions — word by meaningless word — it may be all too easy to forget that the chess pieces are bleeding.