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For a finite-dimensional representation of a complex semisimple Lie algebra, the weights on a line $\mu +\mathbb{Z}\alpha$ form a contiguous string from $\mu +p\alpha$ to $\mu -q\alpha$, with $q-p=⟨\mu ,{\alpha }^{\vee }⟩$. Restriction to the associated sl2 subalgebra gives nested strings of the same parity, so the string endpoints determine the weight set but not its multiplicities.

Every finite-dimensional representation of a complex semisimple Lie algebra is a direct sum of irreducible representations. Equivalently, every invariant subspace has an invariant complement.