The half box is a fundamental domain for the Picard group acting effectively

The closed half box

is a fundamental domain, in the sense of Mathlib's MeasureTheory.IsFundamentalDomain, for the Picard group $\mathrm{SL}_2(\mathbb{Z}[i])$ modulo the kernel $\{\pm 1\}$ of its action on hyperbolic $3$-space. This is the three-dimensional analogue of the modular domain of $\mathrm{SL}_2(\mathbb{Z})$: every orbit meets $B$ (reduction theory for $\mathbb{Z}[i]$: take a point of maximal height on the orbit, translate it into $|x|, |y| \le \tfrac12$, and note that the inversion would raise the height if $|q| < 1$; then fold $y < 0$ onto $y > 0$ by $z \mapsto -z$); an element carrying a point of the open box into the open box acts trivially (comparing heights forces $c = 0$, and the remaining upper triangular element must be $\pm 1$); and the boundary of $B$, contained in four coordinate planes and the unit sphere, has hyperbolic volume zero. Consequently the hyperbolic volume of $B$ is the covolume of $\mathrm{PSL}_2(\mathbb{Z}[i])$.