I use extra strenght Eazy-Off oven cleaner. Smear it on. Wait five minutes max and wipe it off. Completely. Repeat until you get down to bare alloy all over. Wash in warm soapy water thoroughly to ensure that all of the Eazy-Off is off. Then...
Polish. I use a machine polisher that I bought at a yard sale for a few dollars and it works just great, but the machine is not necessary...
For machine polish you will need a couple of grades of polishing compound sticks. The cloth wheels are available at most hardware stores (got mine at Canadian Tire)...
Or, you can go with wet and dry sand paper, starting with 1000 grit, then finer and finer, until the piece shines. Yes, it is a bit of time consuming work, but worth the results, which ever way you go...
Unpolished...
Ten minutes later with machine polish...
Now, add some sunlight and ta-da...