  In this sequel, employing more commutative algebra than that explored in
\cite{CCJ}, we show that an isoparametric hypersurface with four principal
curvatures and multiplicities $(3,4)$ in $S^{15}$ is one constructed by
Ozeki-Takeuchi \cite[I]{OT} and Ferus-Karcher-M\"unzner \cite{FKM}, referred to
collectively as of OT-FKM type.
  In fact, this new approach also gives a considerably simpler, both
structurally and technically, proof \cite{CCJ} that an isoparametric
hypersurface with four principal curvatures in spheres with the multiplicity
constraint $m_2\geq 2m_1-1$ is of OT-FKM type, which left unsettled exactly the
four anomalous multiplicity pairs $(4,5),(3,4),(7,8)$ and $(6,9)$, where the
last three are closely tied, respectively, with the quaternion algebra, the
octonion algebra and the complexified octonion algebra, whereas the first
stands alone by itself in that it cannot be of OT-FKM type.
  A byproduct of this new approach is that we see that Condition B, introduced
by Ozeki and Takeuchi \cite[I]{OT} in their construction of inhomogeneous
isoparametric hypersurfaces, naturally arises.
  The cases for the multiplicity pairs $(4,5),(6,9)$ and $(7,8)$ remain open
now.
