In the algebraic group $G=\operatorname {PCGO}(2m,K)$ where $K$ is an algebraically closed field of an odd characteristic, there is a conjugacy class of involutions with representative: $$ e=\left[ \begin{pmatrix} -wI_m & 0 \\ 0 & wI_m \\ \end{pmatrix} \right]$$ where $w$ is of order 4 in $K$.

The centralizer in $G$ is $C_G(e) =A_{m-1}T_1.2$. The "$2$" acts on $A_{m-1}$ as a graph automorphism and inverts the $T_1$ (one-dimensional torus).

Is there an explicit matrix form of the generator of the "$2$" group?

$G=\operatorname {PCGO}(2m,K)$ is the group which fixes a non-degenerate quadratic from up to a scalar.