  In this paper, we study the commuting variety of symmetric pairs associated
to parabolic subalgebras with abelian unipotent radical in a simple complex Lie
algebra. By using the ``cascade'' construction of Kostant, we construct a
Cartan subspace which in turn provides, in certain cases, useful information on
the centralizers of non $\mathfrak{p}$-regular semisimple elements. In the case
of the rank 2 symmetric pair $(\mathrm{so}_{p+2},\mathrm{so}_{p}\times
\mathrm{so}_{2})$, $p\geq 2$, this allows us to apply induction, in view of
previous results of the authors, and reduce the problem of the irreducibility
of the commuting variety to the consideration of evenness of
$\mathfrak{p}$-distinguished elements. Finally, via the correspondence of
Kostant-Sekiguchi, we check that in this case, $\mathfrak{p}$-distinguished
elements are indeed even, and consequently, the commuting variety is
irreducible.
