Yes, you can omit "already". There are many words you can omit: "if you have not already" (omits "done so") "if you have not" (omits "already done so") "if not done" (omits "you have" and "already") However, your original sentence is superior. The words "already" and "so" add so much clarity.

Yes, you can omit "already" and say, "if you have not done so." The inclusion of "already" adds emphasis to the fact that the action may have already been performed by some individuals, but it is not necessary for the sentence's basic meaning. Yes, "if you have not done so" is equivalent in meaning to "if you haven't done it yet." Both phrases convey the idea that the action (completing the Pretest) has not been accomplished up to this point and should be done now if it hasn't already been done.