  By building on the work in Kuzmak & Tkachuk, "Preparation of quantum states
of two spin-$\frac{1}{2}$ particles in the form of the Schmidt decomposition",
Physics Letters A, {\bf 378}, pp1469-1474, which outlined the control of the
degree of entanglement within this system, it is proven that any $SU(4)$ state
manipulation operator can be realised for this system using a sequence of
pulsed magnetic fields in either two linearly independent directions if the
gyromagnetic ratios are unequal or three directions for equal gyromagnetic
ratios. To achieve this goal, an elementary Lie theoretic proof of the fact
that the group of transformations generated by finite products of exponentials
of a set of Lie algebra vectors is equal to the Lie group generated by the
smallest Lie algebra containing those vectors is rewritten into an explicit
algorithm. A numerical example as well as the proof of the algorithm's
effectiveness is given.
