Affine central coroot from the transpose null ray

Row reduction of the integer matrix $A^t$ uses rational operations. Its rank over $\mathbb{Q}$ equals its rank over $\mathbb{R}$, since the same nonzero minors determine rank. Thus its rational kernel has dimension one: choose the one free coordinate to be $1$ and solve the pivot equations to obtain a nonzero rational null vector. Its real span is the real kernel by F1. Since that kernel contains a strictly positive vector, all its coordinates have one strict sign; change the sign if necessary. Multiply by the product of the finitely many positive denominators to get a positive integer null vector, then divide its entries by their positive gcd to obtain $b$. If $b^{\mathrm{\prime }}$ is another such primitive positive vector, the common null line gives $b^{\mathrm{\prime }}=qb$ with positive rational $q$. Write $q=r\mathrm{/}s$ in lowest positive integer terms. Since every $r{b}_i\mathrm{/}s$ is integral and $r,s$ are coprime, $s$ divides every $b_i$, hence $s=1$. Primitivity of $b^{\mathrm{\prime }}$ then forces $r=1$. This proves uniqueness.

Independence and the embedded Cartan in F2 and F4 make $c_A\mathrm{\ne }0$. For every $j$, F2 and F3 give $[c_A,e_j]=({\sum }_i{b}_i{a}_{ij})e_j=(A^{t}b{)}_j{e}_j=0$ and $[c_A,f_j]=-(A^{t}b{)}_j{f}_j=0$. It also commutes with every Cartan element. The identity $[c_A,[x,y]]=[[c_A,x],y]+[x,[c_A,y]]$ propagates these equalities through all Lie words in the generators, proving centrality. Moreover $c_A={\sum }_i{b}_i[e_i,f_i]$ by F3, so it belongs to $[\mathfrak{g},\mathfrak{g}]$.

On a highest vector $v$ of weight $\lambda$, the Cartan action gives $c_{A}v=\lambda (c_A)v$. By 2.1, commuting $c_A$ through any finite enveloping word $u$ gives $c_{A}uv=u{c}_{A}v=\lambda (c_A)uv$. Such vectors span the cyclic module, proving scalar action. Positive normalization rules out $b=0$, whereas level zero is permitted. A single affine index cannot occur since the one-by-one GCM $[2]$ has no null vector; no empty-index case is hidden in the indecomposable affine hypothesis. The rational elimination, denominator clearing and gcd operations are finite and require no AC.