I'm having trouble understanding connectedness from a group theoretic perspective.
Let $G$ be the symplectic group of dimension 4 over a field $K$,
$G = \operatorname{Sp}_4(K) = \left\{ A \in \operatorname{GL}_4(K) : A^T J A = J \right\} \text{ where } J = \left(\begin{smallmatrix}.&.&.&1\\.&.&1&.\\.&-1&.&.\\-1&.&.&.\end{smallmatrix}\right)$
and let $C$ be the centralizer of a specific unipotent element $t$,
$C=C_G(t) \text{ where } t = \left(\begin{smallmatrix}1&1&.&.\\.&1&.&.\\.&.&1&1\\.&.&.&1\end{smallmatrix}\right)$
The exercise asks one to, Show that $t$ does not lie in the connected component of the identity when the characteristic of $K$ is 2. I think K is algebraically closed, though this is perhaps not specified here (and is specified in a nearby exercise).
I calculate the centralizers to be:
$C_{\operatorname{GL}_4(K)}(t) = \left\{ \left(\begin{smallmatrix}a&b&c&d\\.&a&.&c\\e&f&g&h\\.&e&.&g\end{smallmatrix}\right) : a,b,c,d,e,f,g,h \in K, ag-ec \neq 0 \right\} \cong \operatorname{GL}_2\left(K[dx]/{(dx)}^2\right)$ $C_{\operatorname{Sp}_4(K)}(t) = \left\{ \left(\begin{smallmatrix}a&b&c&d\\.&a&.&c\\e&f&g&h\\.&e&.&g\end{smallmatrix}\right) : a,b,c,d,e,f,g,h \in K, ag-ec = 1, ah+bg=cf+de \right\}$
I am clueless how to find their connected components.
What are the connected components of $C_{\operatorname{GL}_4(K)}(t)$ and $C_{\operatorname{Sp}_4(K)}(t)$?
Especially describe the exceptional behavior in characteristic 2.
Does the connectedness have anything to do with them being matrices?
I would prefer some group theoretic way to find the components, but I worry that the components have nothing to do with the matrices, and depend only on the equations $ag-ec=1$ and $ah+bg=cf+de$, regardless of where these variables are in the matrix.
If they don't have anything to do with the group structure, then why would I care if it is connected?