  In this paper we further develop the theory of $\alpha$-induction for nets of
subfactors, in particular in view of the system of sectors obtained by mixing
the two kinds of induction arising from the two choices of braiding. We
construct a relative braiding between the irreducible subsectors of the two
``chiral'' induced systems, providing a proper braiding on their intersection.
We also express the principal and dual principal graphs of the local subfactors
in terms of the induced sector systems. This extended theory is again applied
to conformal or orbifold embeddings of SU(n) WZW models. A simple formula for
the corresponding modular invariant matrix is established in terms of the two
inductions, and we show that it holds if and only if the sets of irreducible
subsectors of the two chiral induced systems intersect minimally on the set of
marked vertices i.e. on the ``physical spectrum'' of the embedding theory, or
if and only if the canonical endomorphism sector of the conformal or orbifold
inclusion subfactor is in the full induced system. We can prove either
condition for all simple current extensions of SU(n) and many conformal
inclusions, covering in particular all type I modular invariants of SU(2) and
SU(3), and we conjecture that it holds also for any other conformal inclusion
of SU(n) as well. As a by-product of our calculations, the dual principal graph
for the conformal inclusion $SU(3)_5 \subset SU(6)_1$ is computed for the first
time.
