We classify all pairs of reductive maximal connected subgroups of a classical algebraic group
G
G
that have a dense double coset in
G
G
. Using this, we show that for an arbitrary pair
(
H
,
K
)
(H, K)
of reductive subgroups of a reductive group
G
G
satisfying a certain mild technical condition, there is a dense
H
,
K
H, K
-double coset in
G
G
precisely when
G
=
H
K
G = HK
is a factorization.