Geodesic
Existence of Weight Space Decompositions for Irreducible Representations of Simple Lie Algebras
Canadian mathematical bulletin, Tome 14 (1971) no. 1, pp. 113-115

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Let L denote a finite-dimensional simple Lie algebra over an algebraically closed field K of characteristic zero. It is well known that every finite-dimension 1, irreducible representation of L admits a weight space decomposition; moreover every irreducible representation of L having at least one weight space admits a weight space decomposition. 
Lemire, F. W. Existence of Weight Space Decompositions for Irreducible Representations of Simple Lie Algebras. Canadian mathematical bulletin, Tome 14 (1971) no. 1, pp. 113-115. doi: 10.4153/CMB-1971-021-7
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