In the algebraic group $G=\operatorname {PCO}(4,K)$ where $K$ is an algebraically closed field of an odd characteristic, how many different classes of involutions are there in $G \setminus \operatorname{PSO}(4,K)$ and what are the representatives and the centralizer structure in $G$?
I'm aware that there is one class with representative $$e=\left[ \begin{pmatrix} -1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \\ \end{pmatrix} \right]$$
The centralizer in $G$ is $C_G(e) =\operatorname{SO}(3,K) \times 2$, if I'm correct?
Is there any other class?
Any help is appreciated.
$\operatorname{CO}$ is the subgroup of the general linear group which fixes a non-degenerate quadratic form up to a scalar. $\operatorname {PCO}$ is the projective group.