  In the first half of this article, we review the Steinberg theory for double
flag varieties for symmetric pairs. For a special case of the symmetric space
of type AIII, we will consider $ X = GL_{2n}/P_{(n,n)} \times GL_n / B_n^+
\times GL_n / B_n^- $ on which $ K = GL_n \times GL_n $ acts diagonally. We
give a classification of $ K $-orbits in $ X $, and explicit combinatorial
description of the Steinberg maps.
  In the latter half, we develop the theory of embedding of a double flag
variety into a larger one. This embedding is a powerful tool to study different
types of double flag varieties in terms of the known ones. We prove an
embedding theorem of orbits in full generality and give an example of type CI
which is embedded into type AIII.
