Let $\mathbb P_1$ be the one dimensional complex projective space. What is the connected component of the full automorphism of $\mathbb C^*\times \mathbb P_1$. Is it a complex Lie group? I mean is it finite dimensional?

We know that ${\rm Aut}(\mathbb C)^\ast$ is ${\mathbb C}^\ast$, and $Aut^\circ(\mathbb P_1)$ is $PSL(2,\mathbb C)$.