The uniform structure of ${\mathfrak{g}}^{\otimes 4}$

We obtain a uniform decomposition into Casimir eigenspaces (most of which are irreducible) of the fourth power of the adjoint representation ${\mathfrak{g}}^{\otimes 4}$ for all simple Lie algebras. We present universal, in Vogel’s sense, formulae for the dimensions and split Casimir operator’s eigenvalues of all terms in this decomposition. We assume that a similar uniform decomposition into Casimir eigenspaces with universal dimension formulae exists for an arbitrary power of the adjoint representations.