  For finite dimensional hermitean inner product spaces $V$, over $*$-fields
$F$, and in the presence of orthogonal bases providing form elements in the
prime subfield of $F$, we show that quantifier free definable relations in the
subspace lattice $L(V)$ with involution by taking orthogonals, admit quantifier
free descriptions within $F$, also in terms of Grassmann-Pl\"ucker
coordinates.In the latter setting, homogeneous descriptions are obtained if one
allows quantification type $\Sigma_1$. In absence of involution, these results
remain valid.
