Corollary

The group Aut$_e(\mathfrak{g})$ operates transitively on the set of pairs $(\mathfrak{t},\mathfrak{b})$ where $\mathfrak{t}$ is a splitting Cartan subalgebra of $\mathfrak{g}$ and $\mathfrak{b}$ is a Borel subalgebra of $(\mathfrak{g},\mathfrak{t})$.

Let $(\mathfrak{t}_1,\mathfrak{b}_1)$ and $(\mathfrak{t}_2,\mathfrak{b}_2)$ be two such pairs. There exists a splitting Cartan subalgebra $\mathfrak{t}$ of $\mathfrak{g}$ contained in $\mathfrak{b}_1\cap \mathfrak{b}_2$ (Prop. 10). By the Cor. of Prop. 9, we are reduced to the case in which $\mathfrak{t}_1=\mathfrak{t}_2=\mathfrak{t}$. Let S be the root system of $(\mathfrak{g},\mathfrak{t})$. There exists bases $B_1,B_2$ of S such that $\mathfrak{b}_i$ is associated to $B_i(i= 1,2)$, and there exists $s\in W(S)$ which transforms $B_1$ into $B_2$. Finally, there exists $a\in$ Aut$_e(\mathfrak{g})$ such that $a|\mathfrak{t}=s($§2, no. 2, Cor. of Th. 2). Then $a(\mathfrak{t}) =\mathfrak{t}$ and $a(\mathfrak{b}_1) =\mathfrak{b}_2$.