Double cosets of stabilizers of totally isotropic subspaces in a special unitary group I

Let $D$ be a division algebra with a fixed involution and let $V$ be the corresponding unitary space over $D$ with $T$-condition (see [2]). For a pair of totally isotropic subspaces $u,v\leq V$ we consider the double cosets $P_u\gamma P_v$ of their stabilizers $P_u,P_v$ in $\Gamma=\mathrm{SU}(V)$. We give a description of cosets $P_u\gamma P_v$ in the terms of the intersection distance $d_\mathrm{in}(u,\gamma(v))$ and the Witt index of $u+\gamma(v)$.