  In this article we study the construction of characteristic classes for
principal $G$-bundles equipped with an additional structure called
transitionally commutative structure (TC structure). These structures classify,
up to homotopy, possible trivializations of a principal $G$-bundle, such that
the induced cocycle have functions that commute in the intersections of their
domains. We focus mainly on the cases where the structural group $G$ equals
$SU(n)$, $U(n)$ or $\mathrm{Sp}(n)$. Our approach is an algebraic-geometric
construction that relies on the so called power maps defined on the space
$B_{\mathrm{com}}G$, the classifying space for commutativity in the group $G$.
