Split Casimir Operator of Algebra \(D(2,1;\alpha )\) in Representations \(a{{d}^{{ \otimes 2}}}\) and \(a{{d}^{{ \otimes 3}}}\) and Vogel Parameterization

Characteristic identities for \(2\) - and \(3\) -split Casimir algebra operators \(D(2,1;\alpha )\) in representations \(a{{d}^{{ \otimes 2}}}\) and \(a{{d}^{{ \otimes 3}}}\) are found; projectors onto the invariant subspaces of these representations are constructed, and formulas for their supertraces are derived. All formulas agree with the universal description of subrepresentations of representations \(a{{d}^{{ \otimes 2}}}\) and \(a{{d}^{{ \otimes 3}}}\) of basic classical Lie superalgebras in terms of Vogel parameters.