Theorem 2

Let $(\mathfrak{g},\mathfrak{h},B,(X_{\alpha})_{\alpha\in B})$ and $(\mathfrak{g}',\mathfrak{h}',B',(X'_{\alpha})_{\alpha\in B'})$ be framed semi-simple Lie algebras, let R and $R'$ be the root systems of $(\mathfrak{g},\mathfrak{h})$ and $(\mathfrak{g}',\mathfrak{h}')$, let $(n(\alpha , \beta ))_{\alpha ,\beta\in B}$ (resp. $(n'(\alpha , \beta ))_{\alpha ,\beta\in B'}$) be the Cartan matrix of R (resp. $R'$) relative to B (resp. $B'$), and let $\Delta$ (resp. $\Delta '$) be the Dynkin graph of R (resp. $R'$) relative to B (resp. $B'$).

(i) If $\varphi$ is an isomorphism from $\mathfrak{h}^*$ to $\mathfrak{h}^{'*}$ such that $\varphi (R) = R'$ and $\varphi (B) = B'$, there exists a unique isomorphism $\psi$ from $(\mathfrak{g},\mathfrak{h},B,(X_{\alpha})_{\alpha\in B})$ to $(\mathfrak{g}',\mathfrak{h}',B',(X'_{\alpha})_{\alpha\in B'})$ such that $\psi |\mathfrak{h}=^t\varphi^{-1}$.

(ii) If $f$ is a bijection from B to $B'$ such that $n'(f(\alpha ), f(\beta )) =n(\alpha , \beta )$ for all $\alpha , \beta \in$ B, there exists an isomorphism from $(\mathfrak{g},\mathfrak{h},B,(X_{\alpha})_{\alpha\in B})$ to $(\mathfrak{g}',\mathfrak{h}',B',(X'_{\alpha})_{\alpha\in B'})$.

(iii) If there exists an isomorphism from $\Delta$ to $\Delta '$, there exists an isomorphism from $(\mathfrak{g},\mathfrak{h},B,(X_{\alpha})_{\alpha\in B})$ to $(\mathfrak{g}',\mathfrak{h}',B',(X'_{\alpha})_{\alpha\in B'})$.

This follows immediately from Prop. 4 (i) (making use of Chap. VI, §4, no. 2, Prop. 1 for part (iii)).