Continuity mistake: When Columbo takes the whiskey decanter out of the murdered man's liquor cabinet, he puts his cigar in his mouth with his right hand and opens the bottle with the same hand. A second later, the shot changes to a closeup of him sniffing the contents and the cigar is back in his hand. (00:28:55)
Other mistake: Several of the English characters change back and forth between calling Columbo "leftenant" and "lootenant."
Trivia: A clever, subtle hint as to Mason's guilt - Columbo arrives at the animal pound just in time to stop him feeding Laurel and Hardy (the dogs he trained to kill on command) a large amount of chocolate treats. He has planned this murder meticulously, and he would be aware that chocolate is extremely toxic to dogs. The amount he is planning to feed them from the bulging bag of large chocolate balls he is holding would kill both dogs within hours if ingested.
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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