Vogel universality and beyond

For simple Lie algebras we construct characteristic identities for split (polarized) Casimir operators in representations $T\otimes Y_n$ and $T\otimes Y_n^{\mathrm{\prime }}$, where $T$ – defining (minimal fundamental for exceptional Lie algebras) representation, $Y_n$ – n-Cartan powers of the adjoint representations $ad=Y_1$ and Y_n’ – special representations appeared in the Clebsch-Gordan decomposition of symmetric part of $a{d}^{\otimes n}$. By means of these characteristic identities, we derive (for all simple Lie algebras, except ${\mathfrak{e}}_8$) explicit formulae for invariant projectors onto irreducible subrepresentations arose in the decomposition of $T\otimes Y_n$. These projectors and characteristic identities are written in the universal form for all simple Lie algebras (except ${\mathfrak{e}}_8$) in terms of Vogel parameters. Universal formulas for the dimensions of the Casimir subrepresentations appeared in the decompositions of $T\otimes Y_n$ where found.