I did not say that all Islamic Supremacists opposed elections. Nor did I say that Shi'ia Islamic Supremacists adhered to Salafi Takfir. I simply said that the doctrine of Takfir and its analog for Khomeinists sanction activities normally proscribed by Islam for unbelievers due to the dire threat to the faith. This is the basis for any kind of supremacist movement. White supremacists want different laws for blacks than for whites that keep whites superior. Ditto for Islamic supremacists. Also all four groups you mention use terror to advance statecraft, seek the imposition of Islamic law on their subjects (Muslim and non-Muslim) and celebrate a fanatical hatred of Americans and Jews. Yes there are differences. But there are also important similarities you, what's the word, elide.
I'll try to be clearer than I was yesterday: As I understand it, al-Qaeda's interpretation of takfir is so different from anything practiced by Khomeinists, Hamas, or Hezbollah as to be exclusive. While understanding the concept of takfir is essential to understanding al-Qaeda, I don't think it represents the point of commonality between all these groups that Lake insists.
I recognize that there are some similarities among these groups, the use of terrorist violence, the wearing of turbans, the worship of Allah, the eating of hummus. My point is that the differences between them are more significant for U.S. policy than their similarities, and that throwing them together under an "Islamic Supremacist" banner obscures quite a bit more than it reveals about the nature of the ideologies at play here. In the bloggingheads segment, Lake refers to Hamas, Hezbollah, Khomeinist Iran and al-Qaeda collectively as an "8th century violent movement" which "modernity can't accommodate." This is an astonishingly unsophisticated rendering which ignores the dynamic, complex and often contradictory relationship between nationalism and pan-Islamism of the former groups, with whom I think engagement is not only possible but essential, in order to cast them together with the grandiose ultra-fundamentalism of al-Qaeda, with whom I think engagement would be pointless. (I understand the conversational, of-the-cuff nature of the bloggingheads format, so if Lake wants to restate this, I'd like to give him the opportunity.)
President Bush's "war on terror" has, for the last six years, essentially treated the groups mentioned here as if they represented a single front. I'd suggest that this has not worked out well.