  We construct characteristic identities for the split (polarized) Casimir
operators of the simple Lie algebras in adjoint representation. By means of
these characteristic identities, for all simple Lie algebras we derive explicit
formulae for invariant projectors onto irreducible subrepresentations in
T^{\otimes 2} in the case when T is the adjoint representation. These
projectors and characteristic identities are considered from the viewpoint of
the universal description of the simple Lie algebras in terms of the Vogel
parameters.
