Lui trained in the Snake Pit during his time overseas.The only information we have on Lui's backstory is that he lived overseas before returning to Japan. Gabe even mentions that Lui had changed. And with the knowledge of the Snake Pit and their extremely intense training style, perhaps Lui also trained there at some point. It would explain his ruthless demeanor (not to mention Shu acted similar to Lui when he was Red Eye).
Phi and Hyde's father is Theodore GlassA flashback reveals that Phi and Hyde received Revive Phoenix and Dread Hades as gifts from their father when they were little. It was mentioned that the Beys were cherished possessions of their father. And remember what Boa said back in season 2, that Theodore's one true goal was to obtain the ultimate Bey? What if Theodore created those Beys and then later passed them down to his sons?
Lui and Phi battled in the pastIn Phi's debut, Lui got uncharacteristically angry at the mention of Phi's name before declaring that the only one who would defeat Valt was him. Perhaps, Lui encountered Phi at some point before the start of Turbo and lost. A possible explanation for his absence after the Luinor Cup arc was that the two met up again, and Phi destroyed Brutal Luinor.