The Sunni/Shia split goes way back in Islamic history and they do indeed hate each other; witness the Muslim-on-Muslim bombings in Iraq, something that Saddam’s dictatorship at least was able to keep in check. As soon as Islamic leaders feel they’re in control of a region, the war between these two groups breaks out again.
However, I think the leadership can and does put aside their differences when the matter at issue is that of killing or conquering the infidel.
I'll agree with you there.