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By building on the work in Kuzmak & Tkachuk, "Preparation of quantum statesof 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 thedegree of entanglement within this system, it is proven that any $SU(4)$ statemanipulation operator can be realised for this system using a sequence ofpulsed magnetic fields in either two linearly independent directions if thegyromagnetic ratios are unequal or three directions for equal gyromagneticratios. To achieve this goal, an elementary Lie theoretic proof of the factthat the group of transformations generated by finite products of exponentialsof a set of Lie algebra vectors is equal to the Lie group generated by thesmallest Lie algebra containing those vectors is rewritten into an explicitalgorithm. A numerical example as well as the proof of the algorithm'seffectiveness is given.

Fine Entanglement and State Manipulation of Two Spin Coupled Qubits: A Lie Theoretic Overview