How does Jack Worthing's cigarette case betray his identity to Algernon?

Jack Worthing leaves his cigarette case at Algernon's lodging, but Algernon does not know it is his. The case is a gift from Jack's ward, Cecily, and its engraving mentions his real name. At the beginning of the play, Algernon knows Jack only by his town name of Earnest. When Jack claims the cigarette case, he must admit to the ruse.