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Let g be a semisimple Lie algebra over an algebraically closed field of characteristic 0. The set of fixed points of a semisimple inner automorphism of g is a regular reductive subalgebra of maximal rank [1], so it is defined by a subsystem of the root system Φ of g relative to a suitable Cartan subalgebra. The main theorem of the article characterizes the corresponding subsystems of Φ. The second part of the article shows how to compute the fixed point algebras of semisimple outer automorphisms of g. A complete list of all fixed point algebras is then easily obtainable. The results are applied to bounded symmetric domains. References