A K\"ahler structure for the ${\rm PU}(2, 1)$ configuration space of four points in $S^3$

We show that an open subset ${\mathfrak F}_4''$ of the ${\rm PU}(2,1)$ configuration space of four points in $S^3$ is in bijection with an open subset of %with a K\"ahler structure which is inherited from the one of ${\mathfrak H}^{\star}\times\mathbb{R}_{>0}$, where ${\mathfrak H}^\star$ is the affine-rotational group. Since the latter is a Sasakian manifold, the cone ${\mathfrak H}^\star\times\mathbb{R}_{>0}$ is K\"ahler and thus ${\mathfrak F}_4''$ inherits this K\"ahler structure.