To add; we know that any knight can make a knight1, but to make it 'stick' the knighting knight needs to have the appropriate renown, as explained by the greatest knight of all, Ser Barristan "The Bold" Selmy:
[A Dance With Dragons:]
As he watched them at their drills, Ser Barristan pondered raising Tumco and Larraq to knighthood then and there, and mayhaps the Red Lamb too. It required a knight to make a knight, and if something should go awry tonight, dawn might find him dead or in a dungeon. Who would dub his squires then? On the other hand, a young knight’s repute derived at least in part from the honor of the man who conferred knighthood on him. It would do his lads no good at all if it was known that they were given their spurs by a traitor, and might well land them in the dungeon next to him. They deserve better, Ser Barristan decided. Better a long life as a squire than a short one as a soiled knight.
-A Song of Ice and Fire: A Dance With Dragons, Part Two - After the Feast (The Kingbreaker).
**1.** From *A Dance With Dragons*:
“Ser Rolly,” said the big man. “Rolly Duckfield. Any knight can make a knight, and Griff made me. And you, dwarf?”
-A Song of Ice and Fire: A Dance With Dragons, Part Two - After the Feast (Tyrion).