On floor 48, Asuna´s best friend Lizbeth had a blacksmith shop where she makes swords for Asuna and everyone else upon his request.
Kirito visited the shop and tried Lizbeth ´s best master sword, he found it light so he broke it by mistake.

He wanted a special heavy sword that Lizbeth didn´t have the equipment needed for , so they had to go to floor 55 to get the material. On floor 55, Kirito and Lizbeth faced an ice dragon
that pulled them down to a hole that was his nest.
In the whole, Lizbeth started feeling the warmth of Kirito´s heart and hand.
The next morning, Kirito found the material that they were looking for in the nest, but the problem was that they were stuck with no help of teleport stones or even climbing skills!

When the dragon came back to his nest and attacked them, Kirito did his best to save Lizbeth and to go out of the whole by holding into the dragon. When out of the whole, and up in the sky Lizbeth found herself deeply in love with Kirito.

When they were back to floor 48, she made him a very heavy and good sword (Dark Repulser),
but when Asuna came to check on her, she felt that Asuna and Kirito have some feelings for each other,
so she lost her hope in the relation and went crying near a bridge.
Kirito went after her, and said that the only thing they have to care about now is how to win SAO and to return back home and so that was what he promised.