Connectedness of centralizer exercise

I wrote up my findings in this short note. In brief: the exercise is incorrect. Both centralizers are connected. I confirmed with the authors that the intention of the exercise was an exhibition of Springer's 1966 result that a regular element does not lie in the connected component of its centralizer in a few cases, including symplectic groups in characteristic 2. Unfortunately the element given is not regular. The short note contains an example regular element, a calculation of its centralizer, and the decomposition into its connected components.

To find the connected components one has to factor the defining equations. For $C_{GL}$ this is easy: they are irreducible, and the entire centralizer is its own irreducible component. For $C_{Sp}$ this is a bit more challenging as the two polynomials together still define an irreducible system of polynomial equations. The exercise was intended to be easy: for the regular element, the centralizer factors into two obviously irreducible systems.

Since writing up the solution, I have had help proving the centralizer is connected using more elementary methods, mostly using pre-1920s mathematics. However, a few of the details are still too complicated to justify giving here (where I believe one would like a clear and simple explanation for LVK, and a group theoretic explanation for me).

As far as the importance of connectedness in group theory, this remains a mystery. Certainly no-one who helped me over the course of several weeks used the group structure to solve it, though I believe it could be possible to do so.

I think the claim in the OP is mistaken, because of the following : define for $t \in K$,

$$ p(t)= \left(\begin{smallmatrix}1&t&.0&0.\\0.&1&0.&0.\\0.&0.&1&-t\\.0&0.&0.&1\end{smallmatrix}\right) $$

Then $p$ is polynomial and hence continuous for the Zariski topology. As the image of a connected set by a continuous map stays connected, we deduce that $p(K)$ is connected. So $p(O)=I$ and $p(1)=t$ are in the same connected component, (they are in fact "arcwise connected") contrary to the original claim.