Let $K$ be a field of characteristic not $2$ and $R$ a continuous von Neumann regular ring with centre $Z=Z(R)$ isomorphic to $K$. For an example one may assume $R$ is a matrix ring of $n\times n$-matrices with entries in $K$. If ${\rm P}R^\times:=R^\times/Z^\times$ denotes the quotient of the group of invertible elements by the central elements (${\rm PGL}_n(K)$ in the matrix example), every involution $\bar{u}\in {\rm P}R^\times$ is the image of some $u\in R^\times$ satisfying $u^2\in Z$. We call such $\bar{u}$ (and also $u$) projective involutions. Furthermore $u$ is called of the first kind if $u^2=\alpha\in Z$ is a square in $Z^\times$ and of the second kind if it is not.
Is there a way to decide inside ${\rm P}R^\times$ whether a projective involution is of the first or the second kind, that is to tell which kind $\bar{u}$ is of without knowing $u$?
Please note that I am asking for a purely group theoretical classification of projective involutions (e.g. not depending on an action on a projective space).
