Commuting involutions of Lie algebras, commuting varieties, and simple Jordan algebras

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We study certain "σ-commuting varieties" associated with a pair of commuting involutions of a semisimple Lie algebra. The usual commuting variety of and commuting varieties related to one involution are particular cases of our construction. We develop a general theory of σ-commuting varieties and point out some cases, when they have especially good properties. We show that, for a special choice of commuting involutions, the σ-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 × J→ J is equidimensional; while for all other simple Jordan algebras equidimensionality fails.