  We present the universal, in Vogel's sense, expression for the quantum
dimension of Cartan product of an arbitrary number of adjoint and $X_2$
representations of simple Lie algebras. The same formula mysteriously gives
quantum dimensions of some other representations of the same Lie algebra under
permutations of universal parameters. We list these representations for
exceptional algebras and stable versions for classical algebras, when the rank
of the classical algebra is sufficiently large w.r.t. the powers of
representations. We show that universal formulae can have singularities on
Vogel's plane for some algebras and that they give correct answers when
restricted on appropriate lines on Vogel's plane. We note that the same
irreducible 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.
