Envelope of holomorphy of a tube domain over a complex Lie group

According to S. Bochner 6, 7]: IfD =B +iℝ ⁿ is a tube domain in ℂ ⁿ , where B is a domain in ℝ ⁿ , and if (B)\tilde]\tilde B is the convex envelope of B, then any holomorphic function on D extends to the tube domain (D)\tilde] = (B)\tilde] + i\mathbbRⁿ \tilde D = \tilde B + i\mathbb{R}^n , which is a univalent envelope of holomorphy of D. We give a generalization of this result to (nonunivalent) tube domains over a complex Lie group which admit a closed sub-group as a real form. Application: If (V, φ) is a tube domain over ℂ ⁿ and if B is the convex envelope of ϕ(V)∩ℝ ⁿ in ℝ ⁿ , then (V)\tilde] = B + i\mathbbRⁿ \tilde V = B + i\mathbb{R}^n is an envelope of holomorphy of (V, φ).