Universal and homogeneous embeddings of dual polar spaces of rank 3 defined over quadratic alternative division algebras

Suppose O is an alternative division algebra that is quadratic over some subfield K of its center Z(O). Then with (O,K), there is associated a dual polar space. We provide an explicit representation of this dual polar space into a (6n + 7)dimensional projective space over K, where n = dimK(O). We prove that this embedding is the universal one, provided |K| > 2. When O is not an inseparable field extension of K, we show that this universal embedding is the unique polarized one. When O is an inseparable field extension of K, then we determine the minimal full polarized embedding, and show that all homogeneous embeddings are either universal or minimal. We also provide explicit generators of the corresponding projective representations of the little projective group associated with the (dual) polar space.