text: Label the hydrogen spin as , and the carbon spin by . For illustrative purposes, we assume that the equilibrium state only has polarisation on the -spin (in reality, there will also be polarisation on the  spin, with the relative populations determined by the thermal Boltzmann factors). The -coupling Hamiltonian is
  which gives the following evolution
  The non-trivial commutators used to identify the cyclic subspace for  are
  and consequently the next cyclic rotation
  where we used the mixed-product identity , which relates the matrix and Kronecker products for compatible dimensions of , and also the fact that since the two eigenvalues of any of the spin-1/2 operators  are , any of their squares are given by  by the Cayley–Hamilton theorem.
  Note also that the  term is invariant under the -coupling evolution. That is that the term commutes with the Hamiltonian, and in this case, that can be manually confirmed by evaluating the commutator  using the matrix representations of the spin operators.