in fact, american policy in the middle east may seem impossible to unravel from a distance, but it's very easy to understand (and often easy to predict) if you understand the basic principle, which is to maintain as many divisions as possible. it's not divide and conquer, really. it's divide and subdue. the aim is to ensure strong bilateral relations with each and every state that ensures they all remain reliant on america in isolated and unique ways (they don't want multiple states bringing the same concerns at the same time), and then to encourage conflict amongst all the players in such a way that ensures they're all fighting each other, but prevents any from establishing a regional hegemony. so, they may arm both sides in the isis conflict, as they armed both sides in the iran-iraq war. so long as they are constantly fighting each other, and nobody can ever win, they are all equally in the position of accepting american hegemony. it is only when a specific state actor rejects hegemony that a serious conflict erupts to reinforce it. he kind of got to that a little, when he talked of the relationships between egypt, israel, turkey, iran and the saudis.
so, these arrangements appear bizarre - but it's because there's this unquestioned assumption, which murphy did much to perpetuate, that america desires an end to conflict, that they want certain actors to be victorious, that they want stability and peace,etc. it is the opposite that is true. that's why some people take it upon themselves to call america the great satan, after all.
in truth, putting aside colourful language, it's more comparable to the strategy that byzantine rome used against the tribes to their north, which prevented constantinople from being sacked for a good thousand years. i remember reading through that stuff and just being floored by the tactics, and how they consistently squirmed their way out at the last minute, due to finding some way to pit some barbarian against some other barbarian. it is quite truly the case that the romans held out in a castle by the sea for so long because they were constantly outsmarting their enemies, in ways their enemies often didn't even realize.
it's delicate. and it's often brutal. but it's not complicated, if you get it.