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.