Yes, they are so very interested in fighting each other. That’s why Al Qaida attacked buildings in Tehran on 9/11.
I think (hope) you are playing naive for the sake of a one-liner; But I'll bite: who did al qaida attack before 9-11? One answer is the Soviets, after which they did a lot of killing of each other (sunni vs. sunni).
But just ask who is killing who in Iraq and/or why Saddam and the colonial powers had such a heck of a time imposing order in the ME. Or why Iran fought Iraq.