  One-loop scattering amplitudes in string theories involve configuration-space
integrals over genus-one surfaces with coefficients of Kronecker-Eisenstein
series in the integrand. A conjectural genus-one basis of integrands under Fay
identities and integration by parts was recently constructed out of chains of
Kronecker-Eisenstein series. In this work, we decompose a variety of more
general genus-one integrands into the conjectural chain basis. The explicit
form of the expansion coefficients is worked out for infinite families of cases
where the Kronecker-Eisenstein series form cycles. Our results can be used to
simplify multiparticle amplitudes in supersymmetric, heterotic and bosonic
string theories and to investigate loop-level echoes of the field-theory
double-copy structures of string tree-level amplitudes. The multitude of basis
reductions in this work strongly validate the recently proposed chain basis and
stimulate mathematical follow-up studies of more general configuration-space
integrals with additional marked points or at higher genus.
