How to obtain a diffeomorphism between $SL(2;\mathbb{C})$ and $S^{3}×\mathbb{R}^{3}$?

Question

By polar decomposition, we have that every $A \in SL(2;\mathbb{C})$ can be written uniquely as $A = Ue^{X}$, where $U \in SU(2)$ and $X$ is a self-adjoint matrix with trace zero. I already know that $SU(2)$ is diffeomorphic to $S^{3}$. Can someone give me a hint on how to obtain the diffeomorphism?

Thanks in advance!