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 ${\rm Aut}{\mathbb C}^*$ is ${\mathbb C}^*$, and $Aut^\circ{\mathbb P}_1$ is $PSL(2,\mathbb C)$.