Definable relations in finite dimensional subspace lattices with involution. Part II: Quantifier free and homogeneous descriptions

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.