From their perspective, they want a broader war. In their minds they are the carriers of the Black Banners of prophecy of the end times.
They are fanatics. They thrive off of war and conquest. They derive their best satisfaction by fighting non-believers (as they define them). The Great Satan has shown weakness as of late, and they are drawn to that; they want a part of demonstrating that weaknesses.
They would like nothing more than for us to try to redeploy large formations of American ground troops on a field of their choosing. In their mind, that would be best - but they have another path to victory - appear to humble the Great Satan, to gain a great PSYOPS victory by threatening him in to inaction.
Those are their two paths, but we really don't want to follow either one. So far, we seem to be following the early Afghan model (but with better allies in the Kurds) - where we use a few Special Forces and specialized units on the ground to help indigenous ground forces leverage what is our competitive advantage; technology and airpower.
This is the right thing to do - this is the smart thing to do. If we continue to use this model, and adjust as we go, we can give the enemy what they want; greater US involvement, but we should deliver it not quite in the way they want it - and as a result produce our desired effects, not theirs.
By beheading Americans for simply being Americans - IS is calling America out. We should accept the call - but on our terms. If they want this fight, then we should give it to them.
Two things that we can do quickly.
1. Used to great effect in both of our Iraqi conflicts based on their superb use in Vietnam: B-52 carpet bombing.
2. A few MOAB will do nicely as well.
No need for press conferences or other spoken or written responses. Just action.
After the first flush, keep the B-52 and MOAB going now and then to keep them jumpy - but keep a steady pattern of death from the air from other air assets as well. If it is done in coordination with Syria or Iranian proxies in a temporary alliance of convenience - that is fine.
We need to accept that the killing that needs to be done in the land between the Mediterranean and Mesopotamia will need to follow more Game of Thrones rules of allies and partnerships than nice things done in Vienna, Brussels, or Strasbourg.
Do what must be done, but the Islamic State (IS) must be destroyed wholesale and in detail.
It will become more difficult with time to keep at it, especially near the end game (assume that IS is defeated as much as it can be) and the usual suspects get wobbly - but we need to do it, starting sooner more than later.
We need to play another long game. There are thousands of people holding Western passports fighting for IS - like the guy with the English accent who sawed off Foley's head.
I am quite sure that we are working closely with our allied security services to identify them. They need to not be allowed a day of rest until they are killed or captured. They are the most dangerous ones, especially the Anglosphere citizens.
Just as you cannot negotiate with IS, so too can you not treat the individual members of IS as criminals. They are unlawful enemy combatants and should be treated as such. The only mercy that should be shown to them is if they are captured on US soil or they turn themselves in. Otherwise, they should be on the primary kill list. No reason to expend too much effort to capture them. Just kill.
In the end, this is a religious war - even if you don't think it is: if the people who are trying to kill you are part of some religion-based death cult, then you're in a religious war. Accept that and act accordingly.
Their Strategic Center of Gravity is their ideology - but that is a discussion for a different day.
As a final note, can we call it the Long War again?