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Unexpected proportionalities among spin-spin matrix elements in the f shell

By taking advantage of the automorphisms of the group SO(8), as exhibited by a phase reversal in the quark model of the atomic f shell, it is shown that the part z4 of the magnetic spin-spin interaction possesses matrix elements that go beyond the Wigner-Eckart theorem for Racah's group G2 and can always be expressed as a simple product of two factors rather than a sum over products. By means of the spin-quasispin interchange, this result can be carried over to the operator e3 + Ω, which Racah used in his analysis of the Coulomb interaction between the f electrons. The unexpected proportionalities that are such a striking feature of the matrix elements of e3 + Ω can thus be seen as a consequence of the automorphisms of SO(8).