Calculating double-leaf coefficients in the Hecke category

Given a morphism of Abe-Bott-Samelson bimodules, we construct an algorithm for computing its double-leaves coefficients. The key ingredient is a modification of Bruhat upper-triangularity which allows us to use forward substitution to solve for the double-leaf coefficients. We provide a complete implementation of the algorithm in the Sage computer algebra system. We use this to compute the Zamolodchikov 3-color relation in type $H_3$ for balanced realizations. We further simplify the 3-color relation using thick Soergel calculus and express it as a difference of two thick Elias–Williamson diagrams.