I would try full strength Simple Green concentrate on the outside. Spray it on and let it soak for a while, spray again and wait some more, then spray again and wash with a sponge and hot, soapy water. It might take light scrubbing with a no-scratch pad, but even those will leave bright spots on untreated aluminum if you’re not careful. Afterwards, you can apply car wax to the outside to prevent the grey aluminum oxide from continually getting all over your hands and sponges, and it will make new grease spots easier to clean off in the future.
On the inside, isn’t this what most people recommend for an aluminum food-contact surface? I thought this was a sort-of self-anodized surface that prevents aluminum oxides from leaching into your food? I think this is the recommended surface condition for those aluminum moka pots…
