Quasi-orthogonal subalgebras of 4 × 4 matrices

Maximal Abelian quasi-orthogonal subalgebras form a popular research problem. In this paper quasi-orthogonal subalgebras of M4(C) isomorphic to M2(C) are studied. It is proved that if four such subalgebras are given, then their orthogonal complement is always a commutative subalgebra. In particular, five such subalgebras do not exist. A conjecture is made about the maximal number of pairwise quasi-orthogonal subalgebras of M2n(C).