Li Gan

Re Gan ("Regan" in the Crunchyroll translation of the anime) is the Sage of Mirror Lake and a dedicated researcher on magic. He first appears to discuss the world's magic laws with Shiroe, and later is invited by the Enchanter to travel north to Susukino with him.

Appearance
With his disheveled robes and enigmatic appearance, Re Gan sometimes appears to be a shady character. Tetora calls him a "skeleton," a name that he does not seem to particularly mind. Like all Elves, he has characteristically pointy ears.

Personality
Re Gan appears to be mysterious and enigmatic when he first introduce himself to Shiroe and Akatsuki. He holds great interest in studying the laws and mysteries of magic, and is excited by the meeting of three different kinds of scholars present (himself, an Adventurer, and a member of the Kunie clan).

Background
Although he is a mage, his interest lies in 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 the "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, Re Gan sensed the person he had been looking for has arrived while inside his private library.

Shield and Freedom
Re Gan appeared to Shiroe and Akatsuki through a portal at the balcony, though his entrance made them ready their weapons in self-defense. He then introduced himself to the two Adventurers.

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

Gold of the Kunie arc
Re Gan accompanies Shiroe and Naotsugu north to meet with Kinjo, a member of the Kunie clan, to negotiate for a large amount of gold. Despite his enthusiasm over the congregation of three different types of scholars, his hopes are dashed when Kinjo promptly refuses Shiroe's request without much discussion. He then accompanies the two to Susukino, where Shiroe intended to request Silver Sword's cooperation in overcoming a new Raid quest. During this time, Naotsugu and Re Gan wander Susukino in hopes of finding another possible party member. They stumble upon Tetora, who had fallen in the snow, and Re Gan reveals his liking for panties. He casually mocks Tetora's Idol act, much to her displeasure.

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

Equipment
The headpiece he wears grants him insight and memory retention beyond normal capabilities, and is an item handed down upon Sages of Mirror Lake.

Trivia

 * Although volume 7's character page lists his name as "Regan," his name in Japanese is written with a space in the middle, creating "Re Gan."