Mix some mild dishwashing detergent into a quart of hot water. Gently scrub the aluminum with a soft cloth dipped in the soapy water. You can also use a Brillo pad dipped in warm water. Rub the chair's aluminum parts in a back and forth motion, not in circles. This should remove the old clear-coated finish and any dirt, leaving you with plain, unfinished aluminum.
Boil a quart of water and add in 2 tbsp. of vinegar. Dip a soft cloth into the water and clean the aluminum with the vinegar water. You may wish to use a half a lemon dipped in salt instead. Gently rub the aluminum with this salty lemon half (or with a soft cloth dipped in lemon juice and sprinkled with salt).
Apply an aluminum refinishing product that is recommended by professionals for restoring aluminum finishes. Pour some of the refinisher onto a soft cloth and polish the aluminum with it evenly. Follow all use and safety directions!
Let the aluminum dry thoroughly. Once the aluminum is completely dry (this may take several hours), add a second coat of polish. Let dry, and you’re done!