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.
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Align the pillar without the handle to allow liquid to flow to the barrier to complete the trial of the serpent in the hidden city in the head of the serpent in Shadow of the Tomb Raider
Cut the rope. Turn the pillar with the handle so that the liquid can flow from the pillar to the next pillar.

Align the pillar with the handle to allow liquid to flow from it to the other pillar to complete the trial of the serpent in the hidden city in the head of the serpent in Shadow of the Tomb Raider
Go back up to the top and use a rope arrow to release the liquid.

Use a rope arrow to pull the lever to release the liquid to complete the trial of the serpent in the hidden city in the head of the serpent in Shadow of the Tomb Raider
Go up to the serpent head to set the liquid on fire.