  Let $D^{III}_n$ and $\mathscr{S}_n$ be the Cartan domains of type III that
consist of the symmetric $n \times n$ complex matrices $Z$ that satisfy
$Z\overline{Z} < I_n$ and $\mathrm{Im}(Z) > 0$, respectively. For these
domains, we study weighted Bergman spaces and Toeplitz operators acting on
them. We consider the Abelian groups $\mathbb{T}$, $\mathbb{R}_+$ and
$\mathrm{Symm}(n,\mathbb{R})$ (symmetric $n \times n$ real matrices), and their
actions on the Cartan domains of type III. We call the corresponding actions
Abelian Elliptic, Abelian Hyperbolic and Parabolic. The moment maps of these
three actions are computed and functions of them (moment map symbols) are used
to construct commutative $C^*$-algebras generated by Toeplitz operators. This
leads to a natural generalization of known results for the unit disk. We also
compute spectral integral formulas for the Toeplitz operators corresponding to
the Abelian Elliptic and Parabolic cases.
