Which chemical should I use to wash soot precipitate from cast iron?
Depends on your resources and how delicate the object is (is it an antique?).  


You can put it in a furnace or self-cleaning oven to burn-off the soot. Determining the correct temperature for this will be an art.  
You could put it in a furnace under a hydrogen atmosphere.  the hydrogen will react with the carbon to form hydrocarbon gases and it definitely won't oxidize the iron.  You can also use carbon dioxide which will for carbon monoxide with the soot.  You may not have this sort of equipment though.
You can clean the object in an ultrasonicating bath. These are widely available and the best part is that you will not require chemicals to remove the soot.  This method may take a while but it is the most gentle.