tenses - Is "He was arrested because he murdered his neighbour." a simplification of "He was arrested because he had murdered his neighbour."?

I am familiar with tense simplification in subordinate clauses:

He went out after he put on his coat.

instead of had put, because the conjunction after makes it clear that the action of putting on the coat happened earlier, so it is redundant to also indicate it by using the perfect instead of the simple aspect of the tense (past perfect rather than a past simple).

Does the same line of reasoning apply to

He was arrested because he murdered his neighbour.

instead of had murdered, as the conjunction because establishes a reason-result relationship between the main clause and the subordinate clause, and it should go without saying that reason precedes result, or doesn't it? Isn't it obvious that, at the time of the arrest, the murder had already happened?