= Dominant root-lattice highest weights have zero weight
If a <dominant integral weight> $\lambda$ lies in the <root lattice>, its finite-dimensional irreducible <highest-weight representation> contains zero as a weight. First $\lambda$ is a nonnegative integral combination of <simple roots>: a negative part would have negative <inner product> with $\lambda$, contradicting dominance. For any nonzero weight $\mu=\sum m_i\alpha_i$ with $m_i\geq0$, the identity $(\mu,\mu)=\sum m_i(\mu,\alpha_i)>0$ supplies an index with $m_i>0$ and $\langle\mu,\alpha_i^\vee\rangle>0$. By <Injectivity of sl2 lowering above weight zero>, applying the corresponding <lowering operator> produces a nonzero vector of weight $\mu-\alpha_i$. Repetition reaches zero.
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