When did the Byzantine emperor recognize Charlemagne’s imperial title?

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Painting by Albrecht Durer of Charlemagne in ornate coronation robes and crown, holding a sword and orb, with gold embroidery and regal symbols behind him. — Charlemagne Charlemagne, oil on limewood by Albrecht Dürer, 1512; in the collection of the German National Museum, Nürnberg, Germany.

The Byzantine emperor Michael I recognized Charlemagne’s imperial title in 812, following a period of unsuccessful war and negotiation.1

This recognition was given grudgingly and was limited in scope: Charlemagne was acknowledged only as an emperor, rather than as "emperor of the Romans," as the Byzantine rulers maintained their claim to be the sole legitimate successors to the Roman Caesars.1