Li Gan

Ri Gan ("Regan" in the Crunchyroll translation of the anime) is the Sage of Mirror Lake and a dedicated researcher on magic.

Background
Although he is a mage, his interest lies on magical research especially world-level magic. His knowledge is vast, as he is one of a few people who recognized that the entire world of Elder Tale had been altered by magic three times in its history. He named this phenomenon "World Fraction." In the archive, he found that Shiroe first appeared 98 years ago and believed that he must be an archmage based on the frequency and activeness, leading him to contact the guild master of Log Horizon directly to discuss the World Fraction and the Spirit Theory.

An Invitation from Eastal
During the Round Table Alliance's arrival at the Ancient Palace of Eternal Ice, Ri Gan sensed the person he had been looking for has arrived while inside his private library.

Shield and Freedom
Ri Gan appeared to Shiroe and Akatsuki through a portal at the balcony, though his entrance made them ready their weapons in self-defense.

The Two of Us Shall Waltz
A month after the Return of the Goblin King event, Ri Gan confronts Shiroe and praises him on creating a new kind of magic. When asked how he knew about it, Ri Gan admitted seeing the effects of the new magic during the Goblin Conflict.

Gold of the Kunie arc
Ri Gan accompanies Shiroe and Naotsugu north to Susukino, where Shiroe intended to request Silver Sword's cooperation in overcoming a new Raid Boss. During this time, Naotsugu and Ri Gan bond over panties.

After Shiroe and Naotsugu finish their mission, Ri Gan decides to stay in Susukino to finish up his research on the Transport Gates.