  In this paper, we outline a method to compute supersymmetric one-loop
integrands in ten-dimensional SYM theory. It relies on the constructive
interplay between their cubic-graph organization and BRST invariance of the
underlying pure spinor superstring description. The five- and six-point
amplitudes are presented in a manifestly local form where the kinematic
dependence is furnished by BRST-covariant expressions in pure spinor
superspace. At five points, the local kinematic numerators are shown to satisfy
the BCJ duality between color and kinematics leading to supergravity amplitudes
as a byproduct. At six points, the sources of the hexagon anomaly are
identified in superspace as systematic obstructions to BRST invariance. Our
results are expected to reproduce any integrated SYM amplitude in dimensions
$D< 8$.
