  We present the integrand decomposition of multiloop scattering amplitudes in
parallel and orthogonal space-time dimensions, $d=d_\parallel+d_\perp$, being
$d_\parallel$ the dimension of the parallel space spanned by the legs of the
diagrams. When the number $n$ of external legs is $n\le 4$, the corresponding
representation of the multiloop integrals exposes a subset of integration
variables which can be easily integrated away by means of Gegenbauer
polynomials orthogonality condition. By decomposing the integration momenta
along parallel and orthogonal directions, the polynomial division algorithm is
drastically simplified. Moreover, the orthogonality conditions of Gegenbauer
polynomials can be suitably applied to integrate the decomposed integrand,
yielding the systematic annihilation of spurious terms. Consequently, multiloop
amplitudes are expressed in terms of integrals corresponding to irreducible
scalar products of loop momenta and external momenta. We revisit the one-loop
decomposition, which turns out to be controlled by the maximum-cut theorem in
different dimensions, and we discuss the integrand reduction of two-loop planar
and non-planar integrals up to $n=8$ legs, for arbitrary external and internal
kinematics. The proposed algorithm extends to all orders in perturbation
theory.
