This paper provides an improvement of the solution of the unknown input observability problem introduced in [1]. The improvement regards the case when the system is not canonic with respect to its unknown inputs. In this case, [1] introduced a systematic procedure (from now on, the canonization) able to either set the system in canonic form (for which we can compute the observability codistribution as explained in Section 4 of [1]) or, if the system is not canonic, to directly provide the observability codistribution. This procedure determines several functions and is based on their observability. On the other hand, their observability cannot be proven in general. In particular, their observability depends on the experiment (i.e., on the values taken by the unknown inputs). This paper, provides a new procedure to obtain the canonization, which is based on the computation of several functions that are observable independently of the values taken by the unknown inputs (in other words, these functions are observable in accordance with the definition of observability given in [2], which holds in the presence of unknown inputs).