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We present the universal, in Vogel's sense, expression for the quantumdimension of Cartan product of an arbitrary number of adjoint and $X_2$representations of simple Lie algebras. The same formula mysteriously givesquantum dimensions of some other representations of the same Lie algebra underpermutations of universal parameters. We list these representations forexceptional algebras and stable versions for classical algebras, when the rankof the classical algebra is sufficiently large w.r.t. the powers ofrepresentations. We show that universal formulae can have singularities onVogel's plane for some algebras and that they give correct answers whenrestricted on appropriate lines on Vogel's plane. We note that the sameirreducible representation can have several universal formulae for its(quantum) dimension, and discuss the implication of this phenomena on the Cohen- de Man method of calculation of universal formulae.

On universal quantum dimensions of certain two-parameter series of representations