Plot hole: Dr Kepple is finally caught when he rushes to the hiding place of the calibration converter that he hid in the lamp after the murder. Why is it still there? It was a great hiding place initially when the police were searching people and the building in the hours after the murder but Kepple had plenty of opportunities after that to get rid of it. It is the only piece of evidence linking him to the crime and it is in his office so you would think that his main priority would be to dispose of it as soon as possible.
Factual error: Probably an in joke - the wheels of the "airliner" shown landing at LA international airport are those of a B52 bomber. They are very distinctive and look nothing like those on any airliner ever made.
Trivia: Peter Falk's real-life wife appeared in 6 episodes, each time as a different character.
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Answer: In the show, he was already a Lieutenant. A promotion would put him in the next rank up, which for the L.A.P.D. would be Captain. However, some of a Captain's duties would be overseeing other officers and ensuring they're compliant with policies, regulations, and standards. It would also most likely take him out of the field. This is something Columbo has no desire for as he rarely goes to police HQ's. Nor does he show interest in compliance and standards (for example, not going to his semi-annual evaluation at the firing range). However, he could still be assigned to a higher pay grade based on expertise, which is a form of promotion that does not include rank advancement. This would be going from Lieutenant I to Lieutenant II. I don't believe in the show it's ever started what his pay grade is. Although, in s02e01 (I believe) he mentions making $11K a year. Whether or not this was a true statement on his part, if you could find pay scale information for an LAPD Lieutenant in the 70's, it could give you an idea of his pay grade.
Bishop73