Enter the room and take the PHAROS PILLAR from the pedestal.
Stand facing the trapdoor in the floor next to the pedestal, with the pedestal ahead on Lara's right, and press Action to open the trapdoor.
Notice the ladder on the left side of the trapdoor leading down to the room below.
If you want all the kills, enter the hallway opposite the pedestal to draw out a fire wraith. If you avoid this doorway, the wraith won't appear, but you should open the trapdoor regardless.
NOTE: This hall leads to one of the doors on the upper level of the main hall, marked I in the diagram. However, it can only be opened from the outside. Maybe you opened it earlier.
If you don't trigger the wraith, skip down to the next-to-last screenshot on this page.
If you do trigger the wraith, you'll need to find some water to kill it. Run to the right side of the trapdoor, drop back to grab the ladder.
Press and release Action repeatedly to climb down quickly. At the bottom of the ladder, you'll find you're back in the READING ROOM (behind door E).
Roll and run/jump through the door diagonally opposite the ladder.
Follow the hallway around to the ROOM WITH THE SMALL POOL. Jump into the pool and wait there for the fire wraith to extinguish itself.
Climb out of the water and return to the READING ROOM. Climb back up the ladder and through the trapdoor.
Return through the open gate into the small room with the climbing pole.
This time go through the other gate, now on on the right. If it is not already open, it should open as you approach and then close behind you.