  We study certain "\sigma-commuting varieties" associated with a pair of
commuting involutions of a semisimple Lie algebra $\g$. The usual commuting
variety of $\g$ and commuting varieties related to one involution are
particular cases of our construction. We develop a general theory of
\sigma-commuting varieties and point out some cases, when they have especially
good properties. We show that, for a special choice of commuting involutions,
the \sigma-commuting variety is isomorphic to the commuting variety of a simple
Jordan algebra. As a by-product of our theory, we show that if $J$ is the
Jordan algebra of symmetric matrices, then the product map $J \times J\to J$ is
equidimensional; while for all other simple Jordan algebras equidimensionality
fails.
