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At the end of the episode, when the man who killed Palmer looks for Chloe and Derek in the refinery, we can see a shot where Derek gets his hair out of his face and two cuts later we see the exact same scene again.

If Tony and Michelle gave Jack a new identity to "get across the border" at the end of Day 4, why is he working an oil rig in southern California?

Real Time error: It takes about 50 seconds real time between Jack and Derek getting into the helicopter and them flying through the air in the next scene, however later when Jack, Chloe and Derek try to escape in the helicopter, they wait 40 seconds after entering the chopper before getting out because it is taking too long to power up. Surely 40 seconds would have been enough based on the previous scene. Edit: A deleted scene on the Season 5 DVD clears up this error, where Jack calls Diane while waiting for the helicopter to power up.

Shouldn't the Secret Service in the room have kept Palmer away from the window since it wasn't as bulletproof as the windows in the White House?

When Jack kills the assassin at the end of the hour, the scene is identical to when he killed Nina. In both scenes the person Jack is going to kill is on the ground and wounded. The camera is at the eye level of the person and looking up at Jack. In one scene he is avenging his wife, in the other he is avenging Palmer and Michelle.

Chloe is seen here driving the same blue Prius in Los Angeles that she is seen driving in the Season 5 DVD preview...in Chicago. Unless she drove halfway cross the country just to meet with Jack for five minutes, this seems very unlikely she'd have the exact same vehicle.