In the Movie, "The Book of Eli", after getting the book in his hands, why did he (the town's mayor or whatever) not ask Eli for the key on the spot to verify the contents of the book?
I do not know the answer, however I give a logical reason for Carnegie's actions which is the meat of the question
Carnegie (the town's mayor) has been searching for the book for some time. He believes that the Bible is a source of power that will allow him to build more towns and control people. Eli's first encounter with Carnegie, Carnegie considers killing him until he finds out that Eli is a learned man.
Carnegie makes a offer and sweetens the deal with Carnegie's blind concubine Claudia's daughter Solara. However Solara finds out Eli is not easily swayed by offers of the flesh. Carnegie stumbles across the fact that Eli has a Bible (supposedly the last one).
Eli escapes and Solara follows him, Carnegie chases them to the house with the old cannibals. At this point the excitement gets the better of him and he grabs the book without asking about the key. Once he has the book he leaves to go back to town with Solara in tow in which she causes issues. Carnegie is forced to return to town due to low gas.
So logically the answer is either:
Carnegie is so excited to get the book he forgets the key, fiquring the Engineer can open the book. More so he is prevented from returning due to the actions of Solara.