Let and be commuting involutions of a connected reductive algebraic group with . Let
be the corresponding -grading. If , then maps into , and the zero fiber of this bracket is called a -commuting variety. The commuting variety of and commuting varieties related to one involution are particular cases of this construction. We develop a general theory of such varieties and point out some cases, when they have especially good properties. If is a Hermitian symmetric space of tube type, then one can find three conjugate pairwise commuting involutions , , and . In this case, any -commuting variety is isomorphic to the commuting variety of the simple Jordan algebra associated with . As an application, we show that if is the Jordan algebra of symmetric matrices, then the product map is equidimensional, while for all other simple Jordan algebras equidimensionality fails.
Dmitri I. Panyushev. "Commuting involutions of Lie algebras, commuting varieties, and simple Jordan algebras." Algebra Number Theory 7 (6) 1505 - 1534, 2013. https://doi.org/10.2140/ant.2013.7.1505