  We construct characteristic identities for the split (polarized) Casimir
operators of the simple Lie algebras in defining (minimal fundamental) and
adjoint representations. 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 two cases, when T is the
defining and the adjoint representation. In the case when T is the defining
representation, these projectors and the split Casimir operator are used to
explicitly write down invariant solutions of the Yang-Baxter equations. 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.
