Generalized lax Veronesean embeddings of projective spaces

We classify all embeddings θ : PG(n,K) −→ PG(d,F), with d ≥ n(n+3) 2 and K,F skew fields with |K| > 2, such that θ maps the set of points of each line of PG(n,K) to a set of coplanar points of PG(d,F), and such that the image of θ generates PG(d,F). It turns out that d = 12n(n+ 3) and all examples “essentially” arise from a similar “full” embedding θ′ : PG(n,K) −→ PG(d,K) by identifying K with subfields of F and embedding PG(d,K) into PG(d,F) by several ordinary field extensions. These “full” embeddings satisfy one more property and are classified in [4]. They relate to the quadric Veronesean of PG(n,K) in PG(d,K) and its projections from subspaces of PG(d,K) generated by sub-Veroneseans (the points corresponding to subspaces of PG(n,K)), if K is commutative, and to a degenerate analogue of this, if K is noncommutative.