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Helmers theorem for orthogonal Lie algebras

I prove a theorem about the connection of representations of two commuting orthogonal Lie algebras in the complete configuration space of any number of fermions of several kinds inhabiting a common single-kind configuration space. One algebra conserves the total number of fermions and one does not. The theorem is analogous to one concerning symplectic Lie algebras proven by Helmers in 1961. A particular case of the present theorem applies to the algebra O(8) proposed in 1964 by Flowers and Szpikowski as a quasispin algebra in LS coupling. This O(8) algebra has attracted recent attention in discussions of simultaneous isoscalar and isovector pairing. Possible applications of other cases of the theorem are discussed. From it, I derive in a simple manner a relation between the Casimir invariants of the connected representations.