On the generation of some Lie-type geometries

Let $X_n(K)$ be a building of Coxeter type $X_n=A_n$ or $X_n=D_n$ 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 $\mathrm{\Gamma }(K)=G{r}_J(X_n(K))$ be the $J$-Grassmannian of $X_n(K)$. We prove that $\mathrm{\Gamma }(K)$ cannot be generated over any proper sub-division ring $K_0$ of $K$. As a consequence, the generating rank of $\mathrm{\Gamma }(K)$ 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 $G{r}_{1,n}(A_n(K))$ is infinite, although its embedding rank is either $(n+1{)}^2-1$ or $(n+1{)}^2$.