On Stable Cartan Subgroups in Lie Algebras

This paper investigates 𝛤-stable Cartan subgroups in connected Lie groups and their associated Lie algebras. We extend the results of Borel and Mostow on the existence of automorphism-invariant Cartan subalgebras to the group setting. For a real semisimple Lie algebra 𝔤, we prove that there exists a nonidentity automorphism that fixes representatives of all conjugacy classes of Cartan subalgebras. Explicit constructions for classical Lie algebras (𝐴𝑛, 𝐵𝑛, 𝐶𝑛, 𝐷𝑛) are provided. Applications include characterizing stable Cartan subgroups in quotients and normal subgroups.