Dominant lax embeddings of polar spaces

It is known that every lax projective embedding e : Gamma -> PG(V) of a point-line geometry Gamma admits a hull, namely a projective embedding (e) over tilde: Gamma -> PG((V) over tilde) uniquely determined up to isomorphisms by the following property: V and (V) over tilde are defined over the same skewfield, say K, there is morphism of embeddings (f) over tilde : (e) over tilde -> e and, for every embedding e' : Gamma -> PG(V') with V' defined over K, if there is a morphism g : e' - e then a morphism f : e' also exists such that (f) over tilde = gf. If e = (e) over tilde then we say that e is dominant. Clearly, hulls are dominant. Let now Gamma be a non-degenerate polar space of rank n >= 3. We shall prove the following: A lax embedding e: Gamma -> PG(V) is dominant if and only if, for every geometric hyperplane H of Gamma, e(H) spans a hyperplane of PG(V). We shall also give some applications of the above result.