The affine simple root alpha zero is delta minus the highest root

The adjoint module is simple, since its invariant subspaces are ideals. F2 therefore gives a dominant integral highest weight $\theta$ and support in $\theta -Q_{+}$. Its weights by F1 are $0$ and the roots. The top weight cannot be zero: the support contains both a root $\beta$ and $-\beta$, whereas neither pair can both lie in $-Q_{+}$ by F3. Thus $\theta$ is a positive root dominating all roots. Any other root with that property dominates $\theta$ and is dominated by it, so equals it by independence of the simple roots. Write $\theta ={\sum }_i{m}_i{\alpha }_i$; since each ${\alpha }_i\le \theta$, every integer $m_i\ge 1$.

Every finite root can be moved to a dominant root of the same length: if $\beta (h_i)<0$, replace $\beta$ by $s_i\beta =\beta -\beta (h_i){\alpha }_i$. This increases its integer height and stays in the finite root set, so iteration terminates with all pairings nonnegative. For that dominant root $\gamma$, step 1.1 gives $\theta -\gamma ={\sum }_i{n}_i{\alpha }_i$ with $n_i\ge 0$. Dominance of $\theta ,\gamma$ then gives $(\theta ,\theta )-(\gamma ,\gamma )={\sum }_i{n}_i({\alpha }_i,\theta +\gamma )\ge 0.$ Hence $\theta$ has maximal root length and the normalization in F4 is $(\theta ,\theta )=2$.

Choose $[e_{\theta },f_{\theta }]={\theta }^{\vee }$ using F1. Invariance gives $B(e_{\theta },f_{\theta })=2\mathrm{/}(\theta ,\theta )=1$ as in F4. Thus the mode bracket gives $[f_{\theta }\otimes t,e_{\theta }\otimes t^{-1}]=-{\theta }^{\vee }+c=h_0$. The diagonal entry ${\alpha }_0(h_0)$ is $2$; for $i>0$, ${\alpha }_0(h_i)=-\theta (h_i)$ and ${\alpha }_i(h_0)=-{\alpha }_i({\theta }^{\vee })$. They are nonpositive integers by dominance of $\theta$ and crystallographic integrality, and vanish simultaneously since both are positive multiples of $-(\theta ,{\alpha }_i)$. The finite entries already satisfy the GCM axioms.

Put ${\beta }_0=-\theta$ and ${\beta }_i={\alpha }_i$ for $i>0$. The matrix of step 3.1 has entries $2({\beta }_i,{\beta }_j)\mathrm{/}({\beta }_i,{\beta }_i)$. Multiplication of row $i$ by $({\beta }_i,{\beta }_i)\mathrm{/}2$ gives the symmetric Gram matrix of these $\mathrm{\ell }+1$ vectors. It is positive semidefinite of rank $\mathrm{\ell }$, since the finite simple roots are a basis. Its kernel is spanned by $(1,m_1,\dots ,m_{\mathrm{\ell }})$, all entries positive. Each proper principal Gram matrix is positive definite: any dependence would extend to a kernel vector with a zero coordinate, impossible. This is the affine, rather than finite, symmetrizable matrix associated with the highest-root extension; it defines the untwisted affine convention here.

The $h_0,h_1,\dots ,h_{\mathrm{\ell }}$ are independent since only $h_0$ has a nonzero $c$ coordinate; the ${\alpha }_0,{\alpha }_1,\dots ,{\alpha }_{\mathrm{\ell }}$ are independent since only ${\alpha }_0$ is nonzero on $d$. Also $\dim \widehat{\mathfrak{h}}=\mathrm{\ell }+2=2(\mathrm{\ell }+1)-\mathrm{rank}\widehat{A}$. Together with step 3.1 these verify every minimal-realization condition. All root selections and height iterations were finite. Rank one is included: the same formulas give $\left(\begin{array}{cc}2& -2\\ -2& 2\end{array}\right)$.