The affine A1 simple roots and GCM

The finite roots are $\alpha ,-\alpha$, so the highest root is $\theta =\alpha$. F1 gives the displayed data. Since $\delta (c)=\delta (h)=\alpha (c)=0$, we compute ${\alpha }_0(h_0)=0-\alpha (c-h)=2$, ${\alpha }_1(h_0)=\alpha (c-h)=-2$, ${\alpha }_0(h_1)=-\alpha (h)=-2$, and ${\alpha }_1(h_1)=2$. These are all four entries in the stated row-coroot, column-root convention.

By F2, $e_0=f\otimes t$ and $f_0=e\otimes t^{-1}$, with $[e_0,f_0]=c-h=h_0$; the finite generators lie in degree zero. The matrix has rank one since its second row is the negative of its first, and its null vector is $(1,1)$. Accordingly ${\alpha }_0+{\alpha }_1=\delta$ and $h_0+h_1=c$, both nonzero in the full realization. Thus the two off-diagonal double entries describe an affine, not finite rank-one, matrix. All data are explicit and choice-free.