Geodesic
The split $5$-Casimir operator and the structure of
Izvestiya. Mathematics , Tome 89 (2025) no. 1, pp. 15-25

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In the present paper, using the split Casimir operators, we find the decomposition of the antisymmetric part of the fifth power of the adjoint representation $\mathfrak{ad}^{\otimes 5}$. This decomposition contains, in addition to the representations that appeares in the decomposition of $\mathfrak{ad}^{\otimes 4}$, only one new representation of $X_5$. The universal dimension of this representation for exceptional Lie algebras was proposed in [1]. Our decomposition holds for all Lie algebras.
Keywords: adjoint representation, split Casimir operator, Vogel parameters. 
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A. P. Isaev; S. O. Krivonos. The split $5$-Casimir operator and the structure of. Izvestiya. Mathematics , Tome 89 (2025) no. 1, pp. 15-25. http://geodesic.mathdoc.fr/item/IM2_2025_89_1_a1/