The mystery of Sauron's lost finger

[Rilwen Gamgee]

One of the most memorable occurrences in the Battle of Dagorlad: Isildur cutting off Sauron’s finger and claiming the Ring. Now, possibly Tolkien did not want to delve into this matter further, but it brought about much pondering between my sister and I: exactly what happened to Sauron’s finger after it was severed? Did Elrond, on his way leading Isildur up Mt. Doom, stoop to pick it up, put it in his pocket, and bring it back to Rivendell to encase in a glass box for preservation?

My sister and I have looked for information, but have found none. Perhaps we looked in the wrong place? (As we don’t own HoME, we could not search there.) So, as wild as they may be, what are your thoughts?