Algebraic K-theory of varieties $S{L}_{2n}\mathrm{/}S{p}_{2n}$, $E_6\mathrm{/}{F}_4$ and their twisted forms

Let $S{L}_{2n}$, $S{p}_{2n}$, $E_6=G^{sc}(E_6)$, $F_4=G(F_4)$ be simply connected split algebraic groups over an arbitrary field $F$. Algebraic K-theory of affine homogeneous varieties $S{L}_{2n}\mathrm{/}S{p}_{2n}$ and $E_6\mathrm{/}{F}_4$ is computed. Moreover, explicit elements that generate $K_{\ast }(S{L}_{2n}\mathrm{/}S{p}_{2n})$ and $K_{\ast }(E_6\mathrm{/}{F}_4)$ as $K_{\ast }(F)$-algebras are provided. For some twisted forms of these varieties K-theory is also computed.