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I am wondering if there's an adjective that means "without knowing why". Unknowingly seems to mean "without knowing", which is different.
"He killed her unknowingly" doesn't mean the same thing as "He killed her without knowing why (he did it)".
For example:
They kept hurting each other without knowing why (without knowing the cause).
I can suggest some words that are close, but not perfect.
intuitively means without reasoning, but it suggests that the action also comes naturally to the person. I think that this is the closest word that means doing something without knowing why.
unwittingly means without knowledge and without deliberation, but it suggests a degree of accidental behaviour as opposed to being deliberate and just not knowing why. Someone who unwittingly took an item from a store without paying for it may have just had it fall into their bag by accident.
I also like unintentionally in both your examples. Especially because "why" is the cause or the intention underlying an action. But it also means that there may be no intention at all, as opposed to just being unaware of what the intention was. And unfortunately unintentionally doesn't capture the nuance of the difference between "he didn't know why he was motivated to offend the person" and "he didn't know why the person was offended by his [deliberate] actions".
Interestingly, I don't think there's an adverb that means "knowing why" in the affirmative case either.
With a slight change to the wording, your original phrase seems the most natural:
They didn't know why they kept hurting each other.
But if you must have a single word, the closest I can think of would be a form of ignorant:
[Merriam-Webster]
1 a : destitute of knowledge or education
// an ignorant society
also : lacking knowledge or comprehension of the thing specified
// parents ignorant of modern mathematics
1 b : resulting from or showing lack of knowledge or intelligence
// ignorant errors
2 : unaware, uninformed
In theory, you could use it the adverb (they kept hurting each other ignorantly), but that sounds strange.
It would be more natural to say:
They kept hurting each other out of ignorance.
