Let
G be a connected reductive algebraic group over C. We denote by
K = (
Gθ)
0 the identity component of the fixed points of an involutive automorphism
θ of
G. The pair (
G, K) is called a symmetric pair. Let
Q be a parabolic subgroup of
K. We want to find a pair of parabolic subgroups
P1,
P2 of
G such that (i)
P1 ∩
P2 =
Q and (ii)
P1 P2 is dense in
G. The main result of this article states that, for a simple group
G, we can find such a pair if and only if (
G, K) is a Hermitian symmetric pair. The conditions (i) and (ii) imply that the
K-orbit through the origin (
eP1, eP2) of
G/P1 ×
G/P2 is closed and it generates an open dense
G-orbit on the product of partial flag variety. From this point of view, we also give a complete classification of closed
K-orbits on
G/P1 ×
G/P2.
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