  We present a simplified variant of the integrand reduction algorithm for
multiloop scattering amplitudes in $d = 4 - 2\epsilon$ dimensions, which
exploits the decomposition of the integration momenta in parallel and
orthogonal subspaces, $d=d_\parallel+d_\perp$, where $d_\parallel$ is the
dimension of the space spanned by the legs of the diagrams. We discuss the
advantages of a lighter polynomial division algorithm and how the orthogonality
relations for Gegenbauer polynomilas can be suitably used for carrying out the
integration of the irreducible monomials, which eliminates spurious integrals.
Applications to one- and two-loop integrals, for arbitrary kinematics, are
discussed.
