On embeddings of Grassmann graphs in polar Grassmann graphs

We establish that every embedding of a Grassmann graph in a polar Grassmann graph can be reduced to an embedding in a Grassmann graph or to an embedding in the collinearity graph of a polar space. Also, we consider 3-embeddings, i.e. embeddings preserving all distances not greater than 3, of dual polar graphs whose diameter is not less than 3 in polar Grassmann graphs formed by non-maximal singular subspaces. Using the same arguments we show that every such an embedding can be reduced to an embedding in a Grassmann graph.