where did the elves live in lord of the rings
In wild natural places
Elf (Middle-earth)
 These suggested to him that elves were large, dangerous, beautiful, lived in wild natural places, and practised archery. He invented languages for the Elves including Sindarin and Quenya.
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Elf (Middle-earth) [SEP] Introduction
In J. R. R. Tolkien's legendarium, Elves are a fictional race inhabiting Middle-earth in the remote past. Unlike Men and Dwarves, Elves are immortal. They appear in "The Hobbit" and in "The Lord of the Rings", but their history is described more fully in "The Silmarillion". Tolkien derived his Elves from mentions in the ancient poetry and languages of Northern Europe, especially Old English. These suggested to him that elves were large, dangerous, beautiful, lived in wild natural places, and practised archery. He invented languages for the Elves including Sindarin and Quenya. Tolkien-style Elves have become a staple of fantasy literature both in the West and in Japan.