Laura Ingraham, Karl Rove (Credit: Fox News) WATCH: Laura Ingraham and Karl Rove demonstrate depth of GOP divide with testy exchange about Fox News' transportation arrangements How one arrives at Fox News' New York studios apparently says a lot about one's loyalty to the conservative cause

On Fox & Friends Tuesday morning, the divide between the conservative and establishment wings of the Republican Party became evident in the oddest of places -- a conversation between talk radio personality Laura Ingraham and former George W. Bush adviser Karl Rove about the manner in which Fox News transports them to the studio.

It began as a joke about the fact that as a promotion for the Indy 500, Judge Andrew Napolitano was chauffeured to the morning show's set in an actual Formula 1 car, whereas Karl Rove was stranded across town without a ride to the studio, as it was supposed to pick him up too before it broke down.

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"Who's in charge of dispatch at Fox & Friends anyway?" Rove asked, remotely. "Is it you, Doocy? You told me to be downstairs, 7:45 [and] there is a big, black Suburban with my name on it, misspelled of course -- not some Indy car."

"Normally," Rove continued, "I walk over. I'm not a luxurious kind of guy, I normally walk the block-and-a-half between my hotel and Fox, but you sent a big, black Suburban."

At this point, Ingraham interjected that if you "look really closely at the side of that [Formula 1 car sharing the split screen with Rove], you can see a big 'THE GOP ESTABLISHMENT' decal on the side of the car."

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Doocy tried to shut the conversation down, but Rove wanted to get his shot in. "Normally when they send me a car, it's a Camry, and it's got baccarat crystal in it and bottles of brown stuff -- I thought it was your car."

"You're used to that down in Texas," she replied, "the high life."

"Pick-up trucks!" Rove shouted. "Pick-up trucks! And Laura -- I don't drink!"

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"Another reason you're not fun, Karl," she replied. "I'm kidding! I'm kidding!"

"Of course you are," Rove said, and all of the subtle insults were forgotten in a hail of laughter.

"This has really gone off the rails," Doocy said. "To the producers who thought the cross-talk between Laura and Karl would be hilarious? Let's not do this again."

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Watch the entire segment below via Fox News.