BROTHERHOOD OF STEEL QUEST 08 - From Within
Triggered by asking the Brotherhood of Steel to assist you in building the Signal Interceptor during Story 10 - The Molecular Level and talking to Elder Maxson once the Interceptor is complete.
Note: You'll want to wear armour with high energy resistance on upper difficulties, otherwise the lasers will carve you like a ham.
Objective: Locate Doctor Li
After Elder Maxson fills you in about the need to find the doctor, you'll step on the platform and start the events from Story 11 - Institutionalized. Proctor Ingram should also hand over a holodisk for Brotherhood of Steel quest 09 - Outside the Wire before the Interceptor activates. After arriving in the Institute, complete the major story events until you are asked to meet the division leaders. You'll find Doctor Li is the division head of advanced systems (one of the markers from Story 11 - Institutionalized).
Objective: Speak to Doctor Li in Advanced Systems
Talk to her about rejoining the Brotherhood and she'll eventually offer you a deal. You can bypass this if you have high enough charisma and can persuade her. Otherwise it's the evidence on Virgil's death for her help. She will also install a courser chip at this point which allows you to teleport in and out of The Institute after completing Story 11 - Institutionalized.
Objective: Retrieve evidence to convince Doctor Li
You can access the FEV lab via two different doorways in the Bioscience area. If you have the ability hack master level terminals, you can disarm the blue laser barrier at the terminal next to it and simply stroll straight through the door into Virgil's lab. If not, you're going to have to take the novice lock door that can be found down the hallways leading off the Bioscience area.
The way through the FEV lab is extremely linear but quite dangerous. You will be facing quite a few ceiling mounted laser turrets and an assaultron as you work your way through the dingy hallways. You'll also encounter tesla traps and tripwires along the way so stay alert. When you reach a room with super mutants in glass tubes, you've arrived at Virgil's laboratory. The marker on the right will show you the evidence you need to pick up from the table.
Note: You can also pick up Virgil's serum from the device in the corner (you need to remove the tube from it), It's in the back left corner of the room if you are facing into the lab from where you first entered the door.
Objective: Bring the holotape to Doctor Li
Instead of treking back through the FEV area, you can simply take the side door here which will bring you to the other side of the blue laser barrier in the Bioscience area. Interact with the terminal beside it to bring down the grid and you're free to return to Doctor Li. Hand over the evidence to her and she'll hold up her end of the bargain.
Objective: Report to Elder Maxson
As long as you've completed the events of Story 11 - Institutionalized, you should be able to freely fast travel back to the Prydwen to report your success to the elder.