Isometric embeddings of the near polygons and into dual polar spaces

We prove that for every n ∈ N ∖ { 0 , 1 } there exists up to isomorphism a unique isometric embedding of the near polygon H n into the dual polar space D W ( 2 n − 1 , 2 ) and a unique isometric embedding of the near polygon G n into the dual polar space D H ( 2 n − 1 , 4 ) .