Li Gan

ReGan ("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, ReGan 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
ReGan 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 Class 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.

Summer Training arc
During the Round Table Alliance's arrival at the Ancient Palace of Eternal Ice, ReGan sensed the person he had been looking for has arrived while inside his private library. ReGan 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.

They go to ReGan's library where they discuss the World Fraction and the Spirit Theory.

Return of the Goblin King arc
A month after the Return of the Goblin King event, ReGan confronts Shiroe and praises him on creating a new kind of magic. When asked how he knew about it, ReGan admitted seeing the effects of the new magic during the Goblin Conflict.

Gold of the Kunie arc
ReGan 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 ReGan wander Susukino in hopes of finding another possible party member. They stumble upon Tetora, who had fallen in the snow, and ReGan reveals his liking for panties. He casually mocks Tetora's Idol act, much to her displeasure.

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

Hoomsteading The Noosphere
Month's later ReGan returned to Akiba to share his findings with Shiroe, He discovered that the Transport Gate's require large amounts of Mana in order to be powered up and that they are related with the lunar cycle and the supply of that Mana had somehow been cut off.

Shiroe speculated that there might be ancient Alv ruins on the moon and while ReGan could not confirm this, he does speak of legends of Alv's having once resided on the moon in ancient times. while discussing methods to establish communications with the Traveler Faction who are based there, it was ReGan who suggested that he should go to Shibuya as the there is a legend that the old NHK Communications Tower has a method to do just that.

However before they could establish anything further, A crisis occurred involving hundred's of thousands of Moonlight Moths attacking player cities putting all of the People of the Land and Adventurers into a deep sleep, so Shiroe formed the Shibuya Raid Party. ReGan used this opportunity to join them as he wanted to record this unprecedented event that was occurring but also provided crucial assistance as he was the one who was able to unlock the seal to the dungeon raid zone which the Communications tower has now become. While being the weakest member on the team and requiring heavy protection he was able to hold out on his own quite a few times including casting defensive shields against enemy foes.

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."