Properties of finite-type Cartan matrices

$a_{ii}=2({\alpha }_i,{\alpha }_i)\mathrm{/}({\alpha }_i,{\alpha }_i)=2$ and $a_{ij}\in \mathbb{Z}$; for $i\mathrm{\ne }j$ one has $a_{ij}\le 0$ because $({\alpha }_i,{\alpha }_j)\le 0$ by [L2] and $({\alpha }_i,{\alpha }_i)>0$.

$a_{ij}=0$ if and only if $a_{ji}=0$: both entries are nonzero exactly when $({\alpha }_i,{\alpha }_j)\mathrm{\ne }0$, since the denominators $({\alpha }_i,{\alpha }_i),({\alpha }_j,{\alpha }_j)$ are positive.

For $i\mathrm{\ne }j$, the simple roots ${\alpha }_i,{\alpha }_j$ are nonproportional, so $a_{ij}{a}_{ji}=4{\cos }^2\theta \in \{0,1,2,3\}$ by [L3], giving the third assertion.

Let $D=\mathrm{diag}(\mathrm{\mid }{\alpha }_1\mathrm{\mid },\dots ,\mathrm{\mid }{\alpha }_r\mathrm{\mid })$; then $DA{D}^{-1}$ has entries $\mathrm{\mid }{\alpha }_i\mathrm{\mid }{a}_{ij}\mathrm{\mid }{\alpha }_j{\mathrm{\mid }}^{-1}=2({\alpha }_j,{\alpha }_i)\mathrm{/}(\mathrm{\mid }{\alpha }_i\mathrm{\mid }\mathrm{\mid }{\alpha }_j\mathrm{\mid })$, which is symmetric in $i,j$ because the inner product is symmetric.

The matrix in step 1.4 is positive definite: it is twice the Gram matrix $(({\alpha }_i\mathrm{/}\mathrm{\mid }{\alpha }_i\mathrm{\mid },{\alpha }_j\mathrm{/}\mathrm{\mid }{\alpha }_j\mathrm{\mid }))$ of the normalized simple roots, and those vectors are a basis of $E$, so their Gram matrix is positive definite by [L4]. Discarding the factor $2$ preserves positive definiteness.