Embeddings of small generalized polygons

In this paper we consider some finite generalized polygons, defined over a field with characteristic 2, which admit an embedding in a projective or affine space over a field with characteristic unequal to 2. In particular, we classify the (lax) embeddings of the unique generalized quadrangle H(3,4) of order (4,2). We also classify all (lax) embeddings of both the split Cayley hexagon H(2) and its dual H(2)dual in 13-dimensional projective space PG(13,K), for any skew field K. We apply our results to classify the homogeneous embeddings of these small generalized hexagons, and to classify all homogeneous lax embeddings in real spaces of them. Also, we classify all homogeneous embeddings of generalized quadrangles of order (2,2), (4,2) and (2,4).