On the generation of some Lie-type geometries

Let $X_n(K)$ be a building of Coxeter type $X_n=A_n$ or $X_n=D_n$ or defined over a given division ring $K$ (a field when $X_n=D_n$). For a non-connected set $J$ of nodes of the diagram $X_n$, let $\Gamma(K)=Gr_J(X_n(K))$ be the $J$-grassmannian of $X_n(K)$ . We prove that $\Gamma(K)$ cannot be generated over any proper sub-division ring $K_0$ of $K$ . As a consequence, the generating rank of $\Gamma(K)$ is infinite when $K$ is not finitely generated. In particular, if $K$ is the algebraic closure of a finite field of prime order then the generating rank of $Gr_{1,n}(A_n(K))$ is infinite, although its embedding rank is either $(n+1)^2-1$ or $(n+1)^2$.