A good grease remover is Lestoil (good luck, finding that in your grocery store nowadays). Another is ammonia. If you can get commercial grade it is more cost effective and it lasts much longer because it is so strong. Be careful with it; the fumes are overwhelming, and you should dilute it. Be sure the windows are open while you use it. And wear protective gloves, with either of these.
Also, don't use synthetic cloth to clean the stainless steel with either of these products. Use only clean cotton material that has NOT been rinsed with fabric softener. Or use paper towels (but that is so UN-Thrifty).
Your stainless steel will gleam after a wash down with either of these.
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