The split 5-Casimir operator and the structure of $\wedge \ad^{\otimes 5}$

In the present paper, using the split Casimir operators we have found the decomposition of the antisymmetric part of $\ad^{\otimes 5}$. This decomposition contains the representations that appeared in the decomposition of $\ad^{\otimes 4}$ and only one new representation $X_5$. The dimension of this representation has been proposed in [Mac]. Our decomposition is valid for all Lie algebras.