Back in the Archduchy, Michelle went to tell the Archduke about the Soul Seeking Pendants. Cold Flame didn't follow, and instead went to look for Alexander Terce. He had a few questions for him, but just from Michelle's memories, he knew that Alexander was unpredictable and probably wouldn't answer his questions.
Also, from the memories inherited from his ancestors, Cold Flame had a guess about Alexander's identity, which made him frown. Why would a person like that be in the lower realm? And in the Land of Exile at that...
He threw those thoughts behind his head and went to the illegal auction house. If his guess was right, he should be able to meet Alexander if he went to David Elether. And even if that wasn't the case, David could as well answer his questions. If by chance it happened that David wasn't in the auction house, he could just go to the Imperial Palace to search for him since he was at the moment the Prince Regent.