A domain in ${\mathbb{C}}^4$ and its connection with $\mu$-synthesis problem

We explore a domain $\mathbb{F}$ in ${\mathbb{C}}^4$ that has structural similarities with the hexablock $\mathbb{H}\subset {\mathbb{C}}^4$. It leads to the question if these two domains are biholomorphic. In this paper, we answer this question in negative. We provide alternative characterizations of the domain $\mathbb{F}$ and find its connection with the domains associated with $\mu$-synthesis problem such as the symmetrized bidisc, the tetrablock, the pentablock and the hexablock. We also address the following question: does $\mathbb{F}$ (which is biholomorphic with ${\mathbb{L}}_4$) arise from a $\mu$-synthesis problem in the same manner as ${\mathbb{G}}_2$ and $\mathbb{E}$ ?