We introduce a procedure written in the mathematics software suite Maple, which transforms linear time-invariant systems to a special coordinate basis that reveals the internal structure of the system. The procedure creates exact decompositions, based on matrices that contain elements represented by symbolic variables or exact fractions. Throughout the procedure, transformations are constructed with the goal of avoiding unnecessary changes to the original states. The procedure is intended to complement numerical software algorithms developed by others for the same purpose. We discuss various system-theoretic aspects of the special coordinate basis as well as numerical issues related to the decomposition procedure, and illustrate use of the procedure by examples.