Finite generation of division subalgebras and of the group of eigenvalues for commuting derivations or automorphisms of division algebras

Let $D$ be a division algebra such that $D\text{t}{D}^o$ is a Noetherian algebra, then any division subalgebra of $D$ is a {\em finitely generated} division algebra. Let $\text{D}$ be a finite set of commuting derivations or automorphisms of the division algebra $D$, then the group $\text{Ev}(\text{D})$ of common eigenvalues (i.e. {\em weights}) is a {\em finitely generated abelian} group. Typical examples of $D$ are the quotient division algebra $\mathrm{F}\mathrm{r}\mathrm{a}\mathrm{c}(\text{CD}(X))$ of the ring of differential operators $\text{CD}(X)$ on a smooth irreducible affine variety $X$ over a field $K$ of characteristic zero, and the quotient division algebra $\mathrm{F}\mathrm{r}\mathrm{a}\mathrm{c}(U(\text{Gg}))$ of the universal enveloping algebra $U(\text{Gg})$ of a finite dimensional Lie algebra $\text{Gg}$. It is proved that the algebra of differential operators $\text{CD}(X)$ is isomorphic to its opposite algebra $\text{CD}(X{)}^o$.