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Keep it clean, by wiping with a damp paper towel or micro fiber. You can add oil from the back side, if needed. Do so sparingly, just a few drops on a paper towel, then work into the leather.

You...

You do see all kinds of advice on treating leather and a lot depends on the variables, most importantly, what you expect from your strop, do you want it stiff or floppy and the finish you are looking...