A classification is given for regular positions D direct+ D is contained in= D of Jones index 4 where D=alg lim_ -> M_(n_k) (C) is an even matroid algebra and where the individual summands have index 2. A similar classification is obtained for positions of direct sums of 2-symmetric algebras and, in the odd case, for the positions of sums of 2-symmetric C~*-algebras in matroid C~*-algebras. The approach relies on an analysis of intermediate non-self-adjoint operator algebras and the classifications are given in terms of K_0 invariants, partial isometry homology, and scales in the composite invariant K_0(-) direct+ H_1(-).
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