Let
Γ
\Gamma
be a torsion-free, non-uniform lattice in
S
O
(
2
n
,
1
)
\mathrm {SO}(2n,1)
. We present an elementary, combinatorial–geometrical proof of a theorem of Bucher, Burger, and Iozzi [Math. Ann. 381 (2021), pp. 209–242] which states that the volume of a representation
ρ
:
Γ
→
S
O
(
2
n
,
1
)
\rho :\Gamma \to \mathrm {SO}(2n,1)
, properly normalized, is an integer if
n
n
is greater than or equal to
2
2
.