  It was proved by Huckleberry that the Akhiezer-Gindikin domain is included in
the ``Iwasawa domain'' using complex analysis. But we can see that we need no
complex analysis to prove it. In this paper, we generalize the notions of the
Akhiezer-Gindikin domain and the Iwasawa domain for two associated symmetric
subgroups in real Lie groups and prove the inclusion. Moreover, by the symmetry
of two associated symmetric subgroups, we also give a direct proof of the known
fact that the Akhiezer-Gindikin domain is included in all cycle spaces.
