abstract vector space. Rotating the system transforms these states into each other, so this is an example of a "group representation", in this case, the 5-dimensional irreducible representation ("irrep") of the rotation group SU(2) or SO(3), also called the "spin-2 representation". Similarly, the 2p quantum states form a 3-dimensional irrep (called "spin-1"), and the components of the position operator also form the 3-dimensional "spin-1" irrep. Now consider the matrix elements . It turns out that these are transformed by rotations according to the tensor product of those three representations, i.e. the spin-1 representation of the 2p orbitals, the spin-1 representation of the components of r, and the spin-2 representation of the 4d orbitals. This direct product, a 45-dimensional representation of SU(2), is not an irreducible representation, instead it is the direct sum of a spin-4 representation, two spin-3 representations, three spin-2 representations, two spin-1 representations, and a spin-0 (i.e. trivial) representation. The nonzero matrix elements can only come from the spin-0 subspace. The Wigner–Eckart theorem works because the direct product decomposition contains one and only one spin-0 subspace, which implies that all the matrix elements are determined by a single scale factor. Apart from the overall scale factor, calculating the matrix element is equivalent to calculating the projection of the corresponding abstract vector (in 45-dimensional space) onto the spin-0 subspace. The results of this calculation are the Clebsch–Gordan coefficients. The key qualitative aspect of the Clebsch–Gordan decomposition that makes the argument work is that in the decomposition of the tensor product of two irreducible representations, each irreducible representation occurs only once. This allows Schur's lemma to be used. Proof Starting with the definition of a spherical tensor operator, we have which we use to then calculate If we expand the commutator on the LHS by calculating the action of the on the bra and ket, then we get We may combine these two results to get This recursion relation for the matrix elements closely resembles that of the Clebsch–Gordan coefficient. In fact, both are of the form . We therefore have two sets of linear homogeneous equations: one for the Clebsch–Gordan coefficients () and one for the matrix elements (). It is not possible to exactly solve for . We can only say that the ratios are equal, that is or that , where the coefficient of proportionality is independent of the indices. Hence, by comparing recursion relations, we can