Quote 1
"Oh, children," said the Lion, "I feel my strength coming back to me. Oh, children, catch me if you can!" He stood for a second, his eyes very bright, his limbs quivering, lashing himself with his tail. Then he made a leap high over their heads and landed on the other side of the Table. Laughing, though she didn't know why, Lucy scrambled over it to reach him. Aslan leaped again. A mad chase began. (15.40)
Part of Aslan's nature is playfulness. After he is revived by the Emperor's magic, Aslan romps with Susan and Lucy, and all three simply enjoy the feeling of being alive and playing together. There is something simple and beautifully good about their play.
Quote 2
"But where is the fourth?" asked Aslan.
"He has tried to betray them and joined the White Witch, O Aslan," said Mr. Beaver. (12.17-18)
It's interesting that this conversation takes place. Aslan pretty much knows everything that is going on in Narnia, and he definitely knows what's up with Edmund and the Witch, so we assume that asking about the fourth child is just a formality. It needs to be said outright that Edmund is a traitor.
Quote 3
"Hand it to me and kneel, Son of Adam," said Aslan. And when Peter had done so he struck him with the flat of the blade and said, "Rise up, Sir Peter Fenris-Bane." (12.27)
Aslan knights Peter in order to recognize his transition from boy to warrior. The actual knighting ceremony doesn't cause the transformation – it just makes it official.