  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 $\Gamma(K) = Gr_J(X_n(K))$ 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 $(n+1)^2$.
