In the second pair, the subordinate clause starting with 'that' uses a subjunctive. But in the first of the pair 'give' takes two objects '[to] me' and 'the book'. Both sentences in the second pair have two objects that look a bit similar; but 'me' is an indirect object, and in the first pair the 'me' is a direct object.
So in short I don't see any similar rules at work. It's late though, and I'm not at my best (There should be a smiley for :zzz:)
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