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The dynamics-from-permutations of classical Ising spins is studied for achain of four spins. We obtain the Hamiltonian operator which is equivalent tothe unitary permutation matrix that encodes assumed pairwise exchangeinteractions. It is shown how this can be summarized by an exact terminatingBaker-Campbell-Hausdorff formula, which relates the Hamiltonian to a product ofexponentiated two-spin exchange permutations. We briefly comment upon physicalmotivation and implications of this study.

A Baker-Campbell-Hausdorff formula for the logarithm of permutations