It's clear to me that in some situations, "if" works but "whether" does not: 1a) If it rains, I shall take my umbrella. 1b) Call me if rain is predicted.
Also some where only "whether or not" will work: 2a) Whether or not it rains, I still have to go to work.
and some where only "whether" works: 2b) Whether you like this example is of no concern to me.
According to Grammar Girl, there are sitiations where the "or not" is superfluous, as it's not needed to clarify the meaning. 3a) Let me know whether you're going. 3b) Let me know whether or not you're going.
There are also situations where any of the three constructions would work, but the meaning is different:
4a) Let me know if rain is predicted. 4b) Let me know whether rain is predicted. 4c) Let me know whether rain is predicted or not. (to me, 4c above seems ambiguous as to what the speaker wants to know—the prediction or something else altogether— so it needs more context to clarify that)
Can someone succinctly explain which rules of grammar or syntax are operative in these examples, and whether there are other situations where other rules might apply for "if" VS "whether" VS "whether or not"?
Also, are there recognized differences in formal and casual usage?