Automorphism-Invariant Structures in Exceptional and Infinite-Dimensional Lie Algebras

This paper extends the theory of automorphism-invariant Cartan subalgebras to exceptional and infinite-dimensional Lie algebras. Building on Borel-Mostow theory [1] and recent work by Kumar-Mandal-Singh [2], we prove that for any complex exceptional Lie algebra 𝔤 of type 𝐸8 , 𝐹4 , or 𝐺2 , there exists a nonidentity automorphism 𝜎 ∈ 𝐴𝑢𝑡(𝔤) fixing representatives of all conjugacy classes of Cartan subalgebras (Theorem 3.3). For affine Kac-Moody algebras, we establish stability criteria for 𝛤-stable Cartan subalgebras (Theorem 4.4) and construct explicit examples for untwisted affine types. Novel combinatorial invariants are introduced to characterize stability via Dynkin diagram symmetries, and applications to vertex operator algebras are developed. Our results resolve Conjecture 6.1 of Vavilov [4] and Open Problem 1 of Kumar et al. [2]