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It is not a finite dimensional Lie group. For example, all of the maps $$ \Phi\bigl(z,[a,b]\bigr) = \bigl(z, [a+p(z),b]\bigr) $$ where $p:\mathbb{C}^\ast\to \mathbb{C}$ is holomorphic belongs to this group. More generally, the automorphism group is essentially the semi-direct product of $\mathbb{C}^\ast$ with the group of holomorphic mappings $p:\mathbb{C}^\ast\to PSL(2,\mathbb{C})$.