A characterization of the Grassmann embedding of H(q), with q even

In this note, we characterize the Grassmann embedding of H(q), q even, as the unique full embedding of H(q) in PG(12, q) for which each ideal line of H(q) is contained in a plane. In particular, we show that no such embedding exists for H(q), with q odd. As a corollary, we can classify all full solid polarized embeddings of H(q) in PG(12, q); they are necessarily Grassmann embeddings of H(q), with q even. (An embedding is called solid if the lines through a point are contained in a solid.)