  The aim of this paper is to describe the irregular locus of the commuting
variety of a reductive symmetric Lie algebra. More precisely, we want to
enlighten a remark of Popov. In [Po], the irregular locus of the commuting
variety of any reductive Lie algebra is described and its codimension is
computed. This provides a bound for the codimension of the singular locus of
this commuting variety. [Po, Remark 1.13] suggests that the arguments and
methods of [Po] are suitable for obtaining analogous results in the symmetric
setting. We show that some difficulties arise in this case and we obtain some
results on the irregular locus of the component of maximal dimension of the
"symmetric commuting variety". As a by-product, we study some pairs of
commuting elements specific to the symmetric case, that we call rigid pairs.
These pairs allow us to find all symmetric Lie algebras whose commuting variety
is reducible.
