Consider operators  that obey the cyclic commutation relations
In fact only the first two relations are necessary for the following derivation, but since we are usually working with operators associated with Cartesian directions, such as the individual angular momentum operators, the third commutator follows by a symmetry argument.
Introduce also the commutation superoperator  of an operator  (in our case, this is more formally related to the adjoint representation of the Lie algebra whose elements are ), which acts as
In particular, for the cyclic operators, we have
and consequently for integer 
An identity for two operators  is
which can be derived by putting  where  is a scalar parameter, differentiating both sides with respect to , and noting that both sides satisfy the same differential equation in that parameter, with the same initial condition at . In particular, for some scalar parameter , we have
where the final equality follows from recognising the Taylor series for sine and cosine. Now suppose that the density operator at time zero is , and it is allowed to freely evolve under the Hamiltonian  where  is some scalar. Using the results above, the density operator at some later time  will be given by
The interpretation of this is that although nuclear spin angular momentum itself is not connected to rotations in three-dimensional space in the same way that angular momentum is, the evolution of the density operator can be viewed as rotations in an abstract space, in which the operators  are the generators of rotations about the axes. An example of such a set of generators is just the spin operators  themselves.
We now also introduce the arrow notation typically used in NMR, which writes the general evolution given above as the shorthand
With more specific reference to the radiofrequency pulses applied during NMR experiments, a hard pulse with tip angle  around a direction  is written as  above the arrow and corresponds to taking  as the rotation generator in Equation . When there is no ambiguity, the arrow label may be omitted, or be eg text instead.
Note that a more complicated calculation has now been reduced to a simpler procedure that requires no knowledge of the underlying quantum mechanics, especially since the subspaces of cyclic operators can be tabulated in advance.