The basic level one weight of affine sl2

The matrix has diagonal entries $2$, negative off-diagonal entries and a connected two-vertex graph, so it is an indecomposable GCM. Multiplication gives $A(1,1{)}^t=(0,0{)}^t$ and the first row is nonzero while the second is its negative, so the rank is one. F4 makes it affine. Since $A^t=A$, its positive primitive transpose null vector is $(1,1)$, whose entries have gcd one. F1 therefore gives $c_A=h_0+h_1$.

By F5 the Cartan has dimension $4-1=3$ with independent $h_0,h_1$. Fix a complementary vector $z$ so that $(h_0,h_1,z)$ is a basis. For the prescribed $t\in \mathbb{C}$, the formula ${\mathrm{\Lambda }}_0(a{h}_0+b{h}_1+cz)=a+ct$ defines a unique linear functional with the required values. Its labels $1,0$ satisfy F6, independently of $t$.

Evaluating the coroot from 1.1 gives ${\mathrm{\Lambda }}_0(c_A)=1+0=1$. F2 gives integrability of the nonzero simple highest-weight module, and F3 gives infinite dimension. The vanishing second label and arbitrary complementary scalar are both retained, including $t=0$. No zero-level or loop-normalization claim is inferred. All constructions use a finite basis and explicit coordinates, with no AC.