You’ll have to wait around for the elevator before entering.
Once you enter the Cloister proper, go towards the flaming pedestal and retrieve the Kilika Sphere from it. Go to the door and put it on the right-side hole, then remove it again: flames will rise up, then be extinguished, burning away the door.
In the second room, put the Kilika Sphere into the slot on the northern wall, then take it out again and put it on the east wall. Examine the northern glyph to open the door. Head through it.
In this third room, take the Kilika Sphere out from the wall to the east. Go back into the previous room and put it on the west wall, then take the Glyph Sphere from the nearby pedestal. Go north to the larger room and place the Glyph Sphere back to the east, into the same slot from which you grabbed the Kilika Sphere. Backtrack a room and put one of the Kilika Spheres - either is fine - onto the pedestal in the center.
Push the pedestal north into the large room, then east and north onto the glowing floor panel. Go west and downstairs after to the charred floor, then east to find another Kilika Sphere. Snatch it up (hot, hot, hoooot!) and go into the room to the south, placing the Kilika Sphere on the west side of the room. Back in the large room, go east and into the small room that opened up to find a Destruction Sphere within; go downstairs now and place it in the slot on the wall to the east (the same that held the Kilika Sphere moments ago) and you’ll find a chest containing a Red Armlet for Kimahri.
Go back and snatch up just any Kilika Sphere you want, then return to the large room once more. Go downstairs, then up the eastern set of stairs. At the door there, place the Kilika Sphere into the wall, then take it out again to remove the door. Move on ahead. (It doesn’t matter if you take the Sphere with you. Makes you wonder though: if you take it, would other summoners be unable to reach the Fayth? Hm.)
In the room after, you’ll find the rest of your party. Speak with everyone and attempt to leave to make another scene start up.