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From my point of view dwarven hatred towards elves isn’t necessarily based on them being elves, but more based on something the elves at some point did. Lorewise both races have very long lives, which leaves room for petty squabbles easily spanning centuries, since it’s the same people hating each other. In some versions of lore dwarves also carry grudges for ages. TLDR; they don’t hate elves because they are elves but because the elves at some point borrowed the lawnmower and returned it broken or took the last piece of pizza.