  In the present paper, using the split Casimir operators we have found the
decomposition of the antisymmetric part of $\mathfrak{ad}^{\otimes 5}$. This
decomposition contains the representations that appeared in the decomposition
of $\mathfrak{ad}^{\otimes 4}$ and only one new representation $X_5$. The
dimension of this representation has been proposed in [A.J.Macfarlane,
H.Pfeiffer, J.Phys.A: Math. Gen. 36 (2003) 2305]. Our decomposition is valid
for all Lie algebras.
