The insertion algorithm, first introduced by Chang and Lyons (2019), is an efficient solution for inverting the signature of a piecewise linear path. The path is reconstructed by iteratively retrieving its derivative on a specific interval. This (approximated) derivative is the solution of a minimization problem between the signature and the so-called insertion operator, which can be solved very efficiently. We will discuss some modifications of this algorithm and their theoretical grounds. We provide an implementation in Python of this algorithm, to be included in the Signatory package, and show its potential on several examples.