Purity (quest)

Vilkas and/or Farkas wish to be free of the Beast Blood. You are to perform the cleansing ritual that you have done on Kodlak again.

Collect a Glenmoril Witch's head
Should the Dragonborn not have any spare Glenmoril Witch Heads, from playing Blood's Honor, return to Glenmoril Coven and take out and loot the rest of the witches there.

Cleanse the quest giver of beastblood
The quest giver will accompany you to the Burial Chamber in Ysgramor's Tomb. You will need a Glenmoril Witch Head to cure the Beast Blood.

Entering the tomb then take the eastern passage at the statue of Ysgramor to take the shortcut to the Burial Chamber. Wait until the appropriate NPC is standing next to the blue flames, to then toss the head into them. (Otherwise you might accidentally cure yourself.)

Kill the quest giver's wolf spirit
The Spirit of the Beast, a wolf spirit, will appear, kill it.

Talk to the quest giver
Talk the cleansed one. He will be thankful and want to stay in the tomb some more.