  This is the author's second paper treating the double coset problem for
classical groups. Let $G$ be an algebraic group over an algebraically closed
field $K$. The double coset problem consists of classifying the pairs $H,J$ of
closed connected subgroups of $G$ with finitely many $(H,J)$-double cosets in
$G$. The critical setup occurs when one of $H,J$, say $H$, is reductive, and
$J$ is a parabolic subgroup. Assume that $G$ is a classical group, $H$ is
simple and $J$ is a maximal parabolic $P_k$, the stabilizer of a totally
singular $k$-space. Then most candidates have $k=1$ or $k=2$. The case $k=1$
was solved in a previous paper and here we deal with $k=2$. We solve this case
by determining all faithful irreducible self-dual $H$-modules $V$, such that
$H$ has finitely may orbits on totally singular $2$-spaces of $V$.
