Monodromy group for a strongly semistable principal bundle over a curve, II

Let $X$ be a geometrically irreducible smooth projective curve defined over a field $k$. Assume that $X$ has a $k$-rational point; fix a $k$-rational point $x\in X$. From these data we construct an affine group scheme ${\mathcal G}_X$ defined over the field $k$ as well as a principal ${\mathcal G}_X$-bundle $E_{{\mathcal G}_X}$ over the curve $X$. The group scheme ${\mathcal G}_X$ is given by a ${\mathbb Q}$--graded neutral Tannakian category built out of all strongly semistable vector bundles over $X$. The principal bundle $E_{{\mathcal G}_X}$ is tautological. Let $G$ be a linear algebraic group, defined over $k$, that does not admit any nontrivial character which is trivial on the connected component, containing the identity element, of the reduced center of $G$. Let $E_G$ be a strongly semistable principal $G$-bundle over $X$. We associate to $E_G$ a group scheme $M$ defined over $k$, which we call the monodromy group scheme of $E_G$, and a principal $M$-bundle $E_M$ over $X$, which we call the monodromy bundle of $E_G$. The group scheme $M$ is canonically a quotient of ${\mathcal G}_X$, and $E_M$ is the extension of structure group of $E_{{\mathcal G}_X}$. The group scheme $M$ is also canonically embedded in the fiber ${\rm Ad}(E_G)_{x}$ over $x$ of the adjoint bundle.