Northeast of the Akkala Ancient Tech Lab, you can reach Lomei Labyrinth Island, where a mysterious voice gives you a quest.
Quest Location
You can start this quest by checking on the dragon ring in the Lomei Labyrinth Island, which is northeast of the Akkala Ancient Tech Lab in northeast Hyrule.
Lomei Labyrinth Island
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Lomei Sky Labyrinth
The Lomei Sky Labyrinth can be reached from Gikaku Shrine, where you can activate a dragon ring to turn the launcher on, rotate the launcher to point north, then stand on the launcher and glide north to the small island to the north. There you will find a hovercraft that you can fly to the sky labyrinth.
In the sky labyrinth, activate the dragon ring (which can only be done if you have activated the dragon ring on the labyrinth island below), and your quest will be updated. Make sure to activate that quest in your adventure log so you can see the locations of the four terminals. Glide to each of the four terminals by finding gaps in the walls.
After you have activated all four, jump and use the Paraglider to go to the top of the labyrinth, where you can activate the dragon ring. Then you can go forward and press the button that is labeled "Jump" to jump down and skydive to the chasm below. You will likely not be perfectly lined up with the chasm, so be sure to move around as needed to fall through the chasm.
Lomei Depths Labyrinth
In the Lomei Depths Labyrinth, you can go down a gap to find a chest, then use Ascend to go back up to where you started. From there, go down the stairs, throw a Giant Brightbloom Seed if you have one, and defeat the Flux Construct III to complete this quest.
Quest Reward
A chest appears after you defeat the Flux Construct III. Open it to get the Evil Spirit Armor. Afterward, you can go up to the top of this labyrinth to find the Lightroot.