  Associated with a symmetric Clifford system $\{P_0, P_1,\cdots, P_{m}\}$ on
$\mathbb{R}^{2l}$, there is a canonical vector bundle $\eta$ over $S^{l-1}$.
For $m=4$ and $8$, we construct explicitly its characteristic map, and
determine completely when the sphere bundle $S(\eta)$ associated to $\eta$
admits a cross-section. These generalize the results in \cite{St51} and
\cite{Ja58}. As an application, we establish new harmonic representatives of
certain elements in homotopy groups of spheres (cf. \cite{PT97} \cite{PT98}).
By a suitable choice of Clifford system, we construct a metric of non-negative
curvature on $S(\eta)$ which is diffeomorphic to the inhomogeneous focal
submanifold $M_+$ of OT-FKM type isoparametric hypersurfaces with $m=3$.
