Part 1

With the chair upright, raise the seat as high as possible so that the gas cylinder is fully extended.
Turn the chair upside down and place it on a stable, flat surface.
For ease and safety, you are advised to get someone to help you hold the chair steady by gripping the base.
Using the special tool available from your chair supplier, hold it tight to the cylinder and against the chair mechanism (with the cutaway end towards the mechanism and the circular end towards you).  If you do not have the special tool, scroll to the bottom of this page for suggestions.
Strike the tool with a hammer whilst bracing the chair base until the cylinder comes free.

Part 2

Having detached the star base and cylinder from the chair mechanism, turn the star base upside down and support it on the edge of a firm, robust surface (so that the cylinder points downwards and hangs free).
Again, it is advisable to get someone to help you hold it steady.
Use a rubber or wooden mallet to strike the base of the cylinder until it falls out of the star base.
If you do not have a mallet, you can use a hammer.  If you do this, place a piece of wood over the cylinder so that your strike the wood with the hammer (not the metal cylinder base).

A (well used) example of the special removal tool is shown below.  If you do not have access to one of these, you can make one out of wood or metal.  If this is also not an option, you may wish to try hitting the chassis directly with a mallet or hammer but there is a real risk you may damage the chassis if you do this.