To complete the trial of the serpent in the hidden city in the head of the serpent in Shadow of the Tomb Raider use a rope arrow to pull the lever on the wall. Then use the serpent head to burn the blockage. Go down to the two pillars and use a rope arrow to attach the pillar without a handle to the pillar with the handle. Go over to the pillar with the handle and turn it so that the pillar without a handle is turned to allow the liquid to flow to the barrier. Cut the rope. Turn the pillar with the handle so that the liquid can flow from the pillar to the next pillar. Go back up to the top and use a rope arrow to release the liquid. Go up to the serpent head to set the liquid on fire. .
As found on Youtube