Cerium Oxide is the professional way to polish glass. I'm sure it has differing grades of courseness and requires a lot of manual "elbow grease" but it's the best for the job as it's what glaziers use to get a crystal shine.
I'd say from a chemical point of view it'd be pretty inert (don't quote me on this) as it's just an oxidised metal. From my understanding it's like polishing your grandmother's silverware with "Silv-o" or cleaning a kitchen sink with "Jif" you just use a cloth and rub it in circles in the small area until the scratches / blotches are gone.
Polishing glass with Cerium Oxide