Centraliser of an involution in $\text{GL}_{2m}(q)$ Let $m \in \mathbb{N}$ be an integer and $q=2^k$ a power of $2$. Let
$$u = {\rm diag}\left( \begin{pmatrix}
  0 & 1 \\ 1 & 0
  \end{pmatrix}, \cdots, \begin{pmatrix}
  0 & 1 \\ 1 & 0
  \end{pmatrix} \right)$$
be a block diagonal matrix and I am interested in the structure of (or at least the order of) $C_G(u)$ where $G=\text{GL}_{2m}(q)$ is the general linear group over the field with $q$ elements.

If $m=1$, the order is $q(q-1)$. In general, it has the form
$$C_G(u)= \left\lbrace \begin{pmatrix} 
    A_{11} & A_{12} & A_{13} & \cdots \\
    A_{21} & A_{22} & A_{23} & \cdots \\
    \vdots & \vdots & \vdots & \ddots
    \end{pmatrix} \in \text{GL}_{2m}(q) \right\rbrace$$
where the $A_{ij}$ are $2 \times 2$ matrices of the form $\begin{pmatrix}
    a & b \\ b & a
    \end{pmatrix}$
with $a \neq b$, but I don't know how to count the number of such matrices choices with non-zero determinant in general.
 A: With a permutation similarity, $u$ can be rewritten in the form
$$
U = \pmatrix{0 & I_m\\ I_m & 0}.
$$
We see that the matrices that commute with $u$ are those of the form
$$
M = \pmatrix{A & B\\B & A},
$$
with $A,B$ square of size $m \times m$. As is stated here, we have
$$
\det(M) = \det(A - B)\det(A + B) = \det(A + B)^2.
$$
In other words, $M$ is invertible iff $A + B$ is invertible.  That is, for any invertible matrix $P$ and arbitrary matrix $A$, taking $B = P + A$ yields an element of the commutator, and each such element is uniquely specified in this way. Thus, we find that
$$
|C_G(u)| = |C_G(U)| = |GL_m(q)|\cdot q^{m^2}\\
= q^{m^2}(q^m - 1)(q^m - q) \cdots (q^m - q^{m-1}).
$$
A: I find it easier to observe that $u$ is similar to
$$\left(\begin{array}{cc}I_m&I_m \\\\ 0_m&I_m \end{array}\right).$$
Then you can see that the centralizer is
$$\left(\begin{array}{cc}A&B \\\\ 0_m&A \end{array}\right),$$
with $A \in {\rm GL}_m(q)$ and $B$ an arbitrary $m \times m$ matrix.
It has an elementary abelian normal subgroup of order $q^{m^2}$ corresponding to $A=I_m$, with quotient group ${\rm GL}_m(q)$.
