Tony tells Jeannie that 'Iolani Palace in Honolulu is the only royal palace in the United States. However, it is one of two, the other being Hulihe'e Palace in Kailua-Kona, Hawaii.
In the interior scene in the hotel room, after Kamehameha moves offscreen, the reflectors that provide extra light are bumped and the light moves on the wall behind Jeannie.