What word am I looking for?

I don’t think you’re going to do better than “divisive.” You can modify it with “highly” to be more emphatic, but I can’t think of another synonym.

Divisive is derived from “to divide,” which is what you are referring to.

The penguin to the rescue, perhaps.

potentially shattering? explosive? rending?

Polarizing?

polarizing, like it. anything even stronger?

something with fracturing….or that derivative?

Fractious? Contentious?

I agree with @Trillian.

Does it have to be an adjective in front of a noun? If you really want to make it strong, you would do better with a verb:

…the acrimonious debate over Israel and Palestine, which has the power to rupture…

(Avoid using a slash in place of a word. It can stand for more than one word, and you don’t want your reader to have to guess.)

Rancorous? But I most like “polarizing”, as it graphically describes the process of taking polar opposite positions.

If you aren’t only looking for the classy answers given above, I would suggest cluster-fuck.

I like what @CWOTUS suggested. It sounds like dirty fighting. I picture mud slinging, ass kicking, bitch slapping. A very foul disaggrement.
Well at least to me, it paints a picture.

I don’t have any suggestions I’m sorry but I did want to say that I don’t think “polarising” is an overly strong word, not for me anyway… the word “acrimonious” is much stronger, the implication of divisiveness is less direct but still present nonetheless……

Acrimonious means bitter, or caustic. Has nothing to do with being divisive.
@JonnyCeltics you sound like a Hollywood producer; “Yeah, polarizing, I like it. Yeah, what else ya got? Lose the hat, hat’s no good. How ‘bout a bike? Yeah, kid with a bike riding past, whistling. Can you get the kid to whistle? Great, what about the blonde, can we get the blonde in this scene? Yeah great, keep kid, work the blonde in… ya know what? We need the hat after all, put the hat back on. Jerry? Jerry, get me a latte.”

@Trillian nice!

@Trillian” @Jeruba said, ”...the acrimonious debat over Israel and Palestine, which has the power to rupture…” She was using acrimonious to mean bitter or caustic in a phrasal gem..

@gailcalled I was addressing the person above me. I’m all for “acrimonious polarization”. Or “acrimonious divisiveness”. That totally works for me. I was just pointing out that acrimonious alone does not work. It, all by itself, does not imply divisiveness. I also like “schism”.
I would never presume to correct @Jeruba.

Corrosive maybe.

@Trillian: Whew. She certainly is my Supreme Court.

Mine too. I might ask her for clarification if she differs from a point that I had kind of assumed, but I automatically assume she knows more about it than I do. I then assume that I had assumed incorrectly but I need to know what rule is.