Geodesic
Projectors on invariant subspaces of representations $\operatorname{ad}^{\otimes2}$ of Lie algebras $so(N)$ and $sp(2r)$ and Vogel parameterization
Teoretičeskaâ i matematičeskaâ fizika, Tome 206 (2021) no. 1, pp. 3-22

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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.
Keywords: invariant subspace, projector, simple Lie algebra, split Casimir operator, Vogel parameter. 
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     author = {A. P. Isaev and A. A. Provorov},
     title = {Projectors on invariant subspaces of representations $\operatorname{ad}^{\otimes2}$ of {Lie} algebras $so(N)$ and $sp(2r)$ and {Vogel} parameterization},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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     volume = {206},
     number = {1},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2021_206_1_a0/}
}
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A. P. Isaev; A. A. Provorov. Projectors on invariant subspaces of representations $\operatorname{ad}^{\otimes2}$ of Lie algebras $so(N)$ and $sp(2r)$ and Vogel parameterization. Teoretičeskaâ i matematičeskaâ fizika, Tome 206 (2021) no. 1, pp. 3-22. http://geodesic.mathdoc.fr/item/TMF_2021_206_1_a0/