  In this work it is shown that certain interesting types of quasi-orthogonal
system of subalgebras (whose existence cannot be ruled out by the trivial
necessary conditions) cannot exist. In particular, it is proved that there is
no quasi-orthogonal decomposition of M_n(C)\otimes M_n(C)\equiv M_{n^2}(C) into
a number of maximal abelian subalgebras and factors isomorphic to M_n(C) in
which the number of factors would be 1 or 3.
  In addition, some new tools are introduced, too: for example, a quantity
c(A,B), which measures "how close" the subalgebras A,B \subset M_n(C) are to
being quasi-orthogonal. It is shown that in the main cases of interest,
c(A',B') - where A' and B' are the commutants of A and B, respectively - can be
determined by c(A,B) and the dimensions of A and B. The corresponding formula
is used to find some further obstructions regarding quasi-orthogonal systems.
