In the sentence
I have a bibliography page which I'd like to split in/into sections
which would you rather use: split in or split into?
There's not really a "grammatical" justification for the choice, but idiomatically, we almost always use "into" with "sections"...
...whereas with "half", for example (there aren't many such examples), it's the other way around
In fact, apart from "half" I'm not sure there's any other split you can make where "into" isn't preferred. If you compare split in/into two, the preference isn't quite so marked - but that's probably influenced by the unusual usage with "half". As you go to bigger numbers, the preference for "into" is overwhelmingly reasserted.
For a (weak) justification of the idiomatic preference for split into over split in, I suggest that in numerous "compound verbs" (lapse into a coma, descend into chaos, developed into full-blown AIDS, etc.), the "into" component strongly associates with "transformation" of the primary subject.
That association very much involves the "sub-component "to", which can pass muster on its own in things like turn to stone, bring to focus, etc.. With those, you can get away with "into", but you couldn't possibly use "in".
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Split in sections does not make any sense - you would have to split something into sections.
We generally use 'into' to talk about directions and destinations, while we use 'in' to talk about the positions of thing. So you have to use 'split into' in your example because the destinations of the bibliography are the sections. However we might use: (1) 'into' when we think of the division (split) itself; (2) 'in' when we think more at the end of division.
I think either one is acceptable.
The word in has well over 20 definitions, and one of them is into something.
So, I have no problem with divide this page into three sections. Moreover, I would accept divide this page in three sections, although the former seems to sound a bit more natural to me.
As an example of how vexing this could be, I'd probably say: