  We constructed characteristic identities for the 3-split (polarized) Casimir
operators of simple Lie algebras in the adjoint representations $\mathsf{ad}$
and deduced a certain class of subrepresentations in $\mathsf{ad}^{\otimes 3}$.
The projectors onto invariant subspaces for these subrepresentations were
directly constructed from the characteristic identities for the 3-split Casimir
operators. For all simple Lie algebras, universal expressions for the traces of
higher powers of the 3-split Casimir operators were found and dimensions of the
subrepresentations in $\mathsf{ad}^{\otimes 3}$ were calculated. All our
formulas are in agreement with the universal description of (irreducible)
subrepresentations in $\mathsf{ad}^{\otimes 3}$ for simple Lie algebras in
terms of the Vogel parameters.
