BW,

I understand the difference between AQ and HA, but I could just as easily substitute any number of "Pakistani" groups for AQ. The area where AQ has sanctuary is not "ungoverned" as there are local governing structures and peoples which pre-date the creating of the Pakistani state. That "state" of Pakistan has never had control of these areas, nor did the British before them. They are still largely "administered" under the same system used by the British and are quasi-colonies.

Regardless, in Lebanon, I think "Master and Servant" is a poor analogy. HA is a remnant of a civil war, so one can argue that the civil war is still ongoing even though there isn't a lot of fighting anymore. Holding one side in a civil war accountable for the actions of the other side doesn't make a lot of sense to me as the Lebanese government cannot compel/coerce HA.

Now, the US could certainly "incentivize" the Lebanese government and HA to resolve their longstanding differences, but that strikes me as something different from what you're suggesting.