The K-theory of versal flags and cohomological invariants of degree 3

Let $G$ be a split semisimple linear algebraic group over a field and let $X$ be a generic twisted flag variety of $G$. Extending the Hilbert basis techniques to Laurent polynomials over integers we give an explicit presentation of the Grothendieck ring $K_0(X)$ in terms of generators and relations in the case $G=G^{sc}\mathrm{/}{\mu }_2$ is of Dynkin type $\mathrm{A}$ or $\mathrm{C}$ (here $G^{sc}$ is the simply-connected cover of $G$); we compute various groups of (indecomposable, semi-decomposable) cohomological invariants of degree 3, hence, generalizing and extending previous results in this direction.