Pseudoconvexity and Steinness of Connected Complex Lie Groups: A Concise Lie-Theoretic Approach

We give new concise Lie-theoretic proofs of basic analytic–geometric properties of connected complex Lie groups. Using Matsushima’s biholomorphic splitting G≃Cn×K˜ together with a refined analysis of the center via its Cousin factor, we show that every connected complex Lie group is pseudoconvex. Our approach is structural: it reduces to the reductive factor, separates the semisimple and central parts, and concludes using permanence of pseudoconvexity under products and finite quotients, together with standard triviality results for holomorphic principal bundles over Stein bases.