  We generalize arc coordinates for maximal representations from a hyperbolic
surface with boundary into $\text{PSp}(4,\mathbb{R})$, focusing on the case
where the surface is a pair of pants. We introduce geometric parameters within
the space of right-angled hexagons in the Siegel space $\mathcal{X}$. These
parameters enable the visualization of a right-angled hexagon as a polygonal
chain inside the hyperbolic plane $\mathbb{H}^{2}$. We explore the geometric
properties of reflections in $\mathcal{X}$ and introduce the notion of maximal
representation of the reflection group
$W_{3}=\mathbb{Z}/2\mathbb{Z}*\mathbb{Z}/2\mathbb{Z}*\mathbb{Z}/2\mathbb{Z}$.
We parametrize maximal representations from $W_{3}$ into
$\text{PSp}^{\pm}(4,\mathbb{R})$, this induces a natural parametrization of a
subset of maximal and Shilov hyperbolic representations into
$\text{PSp}(4,\mathbb{R})$.
