Turn the chair upside down and locate where the gas lift is fastened to the chair and where it is fastened to the chair's base.
Hit the seat plate with a hammer while using moderate force. The broken gas cylinder and the seat plate are held together by pressure. Each hit with the hammer will release some of that pressure. Make sure to avoid hitting the retainer clip near the top of the gas cylinder. Only make contact with the seat plate.
Pull on the gas cylinder between every several hits to see if it can be pulled out. After several attempts, try grabbing the cylinder with a towel to give you extra grip. Also spray some mechanical lubricant between the cylinder and the seat plate to help loosen the connection if the cylinder is particularly hard to remove.
Flip the base of the chair back into the normal position once the seat has been removed.
Begin hitting the area where the cylinder meets the base of the chair. Again, these two parts are held together by pressure and can be removed by dissipating the pressure with several hits from a hammer. Use your towel and mechanical lubricant if necessary.
Insert a new gas cylinder when the old one has been removed. Close the retaining clip to make sure the pressure locks all the pieces together.
Test the new cylinder by adjusting the lift seat height adjuster. The new cylinder will have sufficient pressure for the height adjuster to work correctly.