Corollary 1

— Let $(\mathfrak g,\mathfrak h,B,(X_\alpha)_{\alpha\in B})$ be a framing of $\mathfrak g$, and $G$ the group (isomorphic to $\operatorname{Aut}(R,B)$) of automorphisms of $\mathfrak g$ that leave this framing invariant. Then $\operatorname{Aut}(\mathfrak g)$ is the semi-direct product of $G$ and $\operatorname{Aut}_0(\mathfrak g)$.

Indeed, every element of $\operatorname{Aut}(\mathfrak g)$ transforms $(\mathfrak g,\mathfrak h,B,(X_\alpha)_{\alpha\in B})$ into a framing of $\mathfrak g$. By Prop. 5, every coset of $\operatorname{Aut}(\mathfrak g)$ modulo $\operatorname{Aut}_0(\mathfrak g)$ meets $G$ in exactly one point.

Q.E.D.

It follows from Cor. 1 that the group Aut($\mathfrak{g}$)$/$Aut$_0(\mathfrak{g})$ can be identified with Aut(R$,B$), and is isomorphic to the group of automorphisms of the Dynkin graph of R.