Projectors on invariant subspaces of representations $\operatorname{ad}^{\otimes2}$ of Lie algebras $so(N)$ and $sp(2r)$ and Vogel parameterization

Using the split Casimir operator, we find explicit formulas for the projectors onto invariant subspaces of the $\operatorname{ad}^{\otimes2}$ representation of the algebras $so(N)$ and $sp(2r)$. We also consider these projectors from the standpoint of the universal description of complex simple Lie algebras using the Vogel parameterization.