No.
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SnarfSo to be an independent clause, it would have to be, "Because they made me angry, that's why"?Not quite. Though technically a fragment, because they made me angry is an elliptical sentence:
Aspara GusNot quite. Though technically a fragment, because they made me angry is an elliptical sentenceMeaning it is a complete sentence?
Aspara GusSo adding an independent clause to it looks too much like a comma spliceI completely agree. Indeed it does.
SnarfMeaning it is a complete sentence?Kinda sorta, but not really.
SnarfOkay, so basically the parts in brackets are implied, just like the "you" in "come on" is implied.No, they are not comparable. Imperative sentences are not elliptical.
SnarfSo then (without removing "because"), rather than having that seeming comma splice when adding "that's why," would it be better to just treat them both like independent clauses in that person's reply (we are, of course, speaking of fictional dialogue right now)?Yes.