I wanted to add my thoughts. Due to finish, I would also be careful of abrasives. They can work, but if they take off the anodized coating, the grill will require repolishing more frequently. What has worked well for me is the Mother's car polish. It is sold in the same section as with regular car wax. Is in intended for use on paint to remove very fine scratches before applying the car wax, and also works well on other finishes. I used it on my TR3A grill that is different from a TR4A grill except that they are both anodized coated aluminum. The grill was 10 years old and using the polish really helped improve the shine. This polish provides a better shine vs regular car wax without removing the finish.
By the way, I also use it on my aluminum rims on my motorcycle that also has a protective coding.