In this work we introduce the concept of Cartan-Polyakov loops, a special subset of Polyakov loops in the fundamental and antifundamental representation of the $\mathrm{SU}(N_c)$ group, ${\ell }_{\mathrm{F},k}$ and ${\bar{\ell }}_{\mathrm{F},k}$ respectively, with charges $k=1,\dots ,(N_c-1)/2$. It constitutes a sufficient set of independent degrees of freedom and it is used to parametrize the thermal Wilson line. Polyakov loops not contained in this set are classified as non-Cartan-Polyakov loops. Using properties of the characteristic polynomial of the thermal Wilson line, we write a non-Cartan-Polyakov loop charge decomposition formula. This formalism allows one to readily build effective models of quarks and gluons with an arbitrary number of colors. We apply it to the Polyakov-Nambu-Jona-Lasinio model and to an effective glue model, in the mean field approximation, showing how to directly extend these models to higher values of $N_c$.