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For a complex semisimple Lie algebra g with Hermitian real form g R = k R + p R , there exists a positive system of roots such that the adjoint k-representation on p stabilizes the positive and negative root spaces. In this article, we extend this result to contragredient Lie superalgebras g, and study the number of irreducible components of the k-representation. We also discuss the complex structure on g R /k R .

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We add extra data to the affine Dynkin diagrams to classify all the finite order automorphisms on real simple Lie algebras. As applications, we study the extensions of automorphisms on the maximal compact subalgebras and also study the fixed point sets of automorphisms.