Animalists hold that we are animals. It is widely accepted by animalists that if a brain were removed from a body, and the body kept alive, the person would stay with the bulk of the body rather than go with the brain.
I wonder how much of the intuition is based on irrelevant questions of physical bulk. Imagine aliens who are giant brains with tiny support organs—lungs, heart, legs, etc.—dwarfed by the brain. I think we might have the intuition that if the brain were disconnected from the support organs, the animal would go with the brain. In the case of beings that dwarf their brains, it feels natural to talk of a certain operation as a brain transplant. But in the case of beings that are almost all brain, the analogous operation would probably be referred to as a support-system transplant. Yet surely we should say exactly the same thing metaphysically about us and the aliens, assuming that the functional roles of the brains and the other organs are sufficiently similar.
This isn't a positive argument that we'd go with our brains. It's just an argument to defuse the intuition that we wouldn't.
What about cerebra? Here's a widely shared intuition. If the cerebrum is removed from the skull of an animal and placed in a life-support vat, the animal stays with the rest of the body.
But now suppose that we granted that the animal goes with the whole brain. Let's say, then, that I am an animal and sadly become a brain in a life-support vat, losing the rest of my body. Suppose that next my brain is cut and the upper and lower brains are placed in separate life-support vats. It does not seem particularly plausible to think that the animal goes with the lower brain. (Maybe the animal dies, or maybe it goes with the upper brain.) So once we've granted that the animal would go with the brain, the primacy of the lower brain for animal identity seems somewhat undermined.
Maybe, though, one could accept both (a) the common intuition that if the cerebrum were removed the human animal would go with the rest of its body, and (b) my intuition that if the human animal were first reduced to a brain, and the brain then cut into the cerebrum and lower brain, the animal would go with the cerebrum. There is no logical contradiction between these two intuitions. Compare this. I have a loaf of bread. Imagine the loaf marked off into five equally sized segments A, B, C, D and E. If I first cut off the 2/5 of the loaf marked D and E, it's plausible that the loaf shrinks to the ABC part, and DE is a new thing. And then if I cut off C, the same loaf shrinks once again, to AB. On the other hand if I start off by cutting off the AB chunk, the loaf shrinks to CDE. So the order of cutting determines whether the original loaf ends up being identical to AB or to something else. (We can also make a similar example using some plant or fungus if we prefer a living example.) Likewise, the order of cutting could determine whether the animal ends up being just a cerebrum (first remove brain, then cut brain into upper and lower parts) or whether it ends up being a cerebrumless body.
We might have a rough general principle: The animal when cut in two tends to go with the functionally more important part. Thus, perhaps, when the human animal is cut into a brain and a rest-of-body, it goes with the brain, as the brain is functionally more important in the brainier animals. When that brain is subsequently cut into upper and lower brains, the brainy animal goes with the upper brain, as that's functionally more important given its distinctively brainy methods for survival. On the other hand, if the human animal is cut into a cerebrum and a cerebrumless-rest-of-body, perhaps (I am actually far from sure about this) the animal goes with the cerebrumless-rest-of-body, because although the upper brain is more important functionally than a lower brain, the lower brain plus the rest of the body are collectively more important than the upper brain by itself. So the order of surgery matters to identity.