It is expressed that the Brotherhood of Steel considers Virgil to be a threat, even after his help in infiltrating the Institute. Eliminate him, and prove the Sole Survivor's loyalty to the war effort.
Lancer-Captain Kells will reveal that Virgil is considered a priority target for assassination, even after his help towards the BoS cause, and his death is a priority mission now for the Sole Survivor.
Upon reaching the rocky cave, the Sole Survivor may choose from a myriad of options in how they want to deal with Virgil:
Should Virgil already be dead, this quest can be immediately completed by informing Kells.
The Sole Survivor may simply kill Virgil, which can be made relatively easier by curing him with the experimental serum. His turrets and protectron guardians must also be dealt with.
Should the Sole Survivor lie to Virgil and tell him that the serum was destroyed, he can be convinced to go down without a fight, which also includes his guardians.
The Sole Survivor may choose to lie for Virgil, in which he will remain in the rocky case.
After reporting back to Lancer-Captain Kells, the Sole Survivor will be rewarded with a unique t-60c helm.