Are the following boldfaced options okay? If any one of them is not okay, could you explain why?
The judge sentenced the murderer to death on December 15 last year, and he was executed in four months / after four months / four months later.
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I would use "four months later" here. "After four months" is ok if you have a bit of context. For example, he was sentenced to death. After four months of failed appeals, he was executed.
It's not entirely incorrect here to say "in four months." It's what we use with future tense: I will be starting my new job in four months (or "after four months of wandering around the world, I will be starting my new job"). But we also use "in" in the past to denote the time an entire process took: I was able to finish the course in three months. (This is why Elainor says it could mean that the execution took four months, and why it isn't quite as accurate in your context.)
Another example: My wife had a major back operation. At her first appointment, the surgeon explained the operation to her, and said to think about it. She said she didn't need to think about it, and to go ahead and set it up. Two weeks later, they performed the operation. I was impressed. They got from the commitment to actually performing the operation in just two weeks.
"In four months", means that he was executed within the four months or that it took four months to execute him. Wouldn't really use this here unless you were talking about someone dying slowly.
"After four months" doesn't make sense with this context. It means that he died after four months. That leads the reader to ask, "then what month was he executed in?"
"Four months later", is more correct because it is specific. Since the date of his sentencing is mentioned on a specific date, then the next part of the sentence should be somewhat specific. The 'later' part implies that his execution was four months from the time of his execution, rather than during four months.