Matsumoto's theorem is the statement that the set of all reduced words for an element of a Coxeter group is spanned and preserved by the group's so-called "braid relations." Hu and Zhang have recently proved an analogue of this classical theorem for certain variants of reduced words for involutions in classical Weyl groups. Hu and Zhang's proof strategy would apply to more general Coxeter groups, but in practice is limited by the formidable amount of algebra involved. In this talk, I will describe an algorithm which passes this technical burden onto a computer and lets us generalize Hu and Zhang's theorem to all finite and affine Coxeter groups. Perhaps more interesting than the output of this algorithm is its implementation, which will highlight a variety of use cases for Sage.