  We compute a class of diagrams contributing to the multi-leg soft anomalous
dimension through three loops, by renormalizing a product of semi-infinite
non-lightlike Wilson lines in dimensional regularization. Using non-Abelian
exponentiation we directly compute contributions to the exponent in terms of
webs. We develop a general strategy to compute webs with multiple gluon
exchanges between Wilson lines in configuration space, and explore their
analytic structure in terms of $\alpha_{ij}$, the exponential of the Minkowski
cusp angle formed between the lines $i$ and $j$. We show that beyond the
obvious inversion symmetry $\alpha_{ij}\to 1/\alpha_{ij}$, at the level of the
symbol the result also admits a crossing symmetry $\alpha_{ij}\to
-\alpha_{ij}$, relating spacelike and timelike kinematics, and hence argue that
in this class of webs the symbol alphabet is restricted to $\alpha_{ij}$ and
$1-\alpha_{ij}^2$. We carry out the calculation up to three gluons connecting
four Wilson lines, finding that the contributions to the soft anomalous
dimension are remarkably simple: they involve pure functions of uniform weight,
which are written as a sum of products of polylogarithms, each depending on a
single cusp angle. We conjecture that this type of factorization extends to all
multiple-gluon-exchange contributions to the anomalous dimension.
