  Let G be a compact connected Lie group and p : E \to {\Sigma}^2V a principal
G-bundle with a characteristic map \alpha : A={\Sigma}V \to G. By combining
cone decomposition arguments in Iwase-Mimura-Nishimoto [3,5] with computations
of higher Hopf invariants introduced in Iwase [8], we generalize the result in
Iwase-Mimura [12]: Let {F_{i}|0 \leq i \leq m} be a cone-decomposition of G
with a canonical structure map \sigma_{i} of cat(F_{i}) \leq i for i \leq m. We
have cat(E) \leq \Max(m+n,m+2) for n \geq 1, if \alpha is compressible into
F_{n} \subseteq F_{m} \simeq G and H^{\sigma_n}_n(\alpha) = 0, under a suitable
compatibility condition. On the other hand, calculations of Hamanaka-Kono [3]
and Ishitoya-Kono-Toda [5] on spinor groups yields a lower estimate for the L-S
category of spinor groups by means of a new computable invariant
Mwgt(-;{mathbb{F}_2}) which is stronger than wgt(-;{\mathbb{F}_2}) introduced
in Rudyak [16] and Strom [18]. As a result, we obtain cat(Spin(9)) =
Mwgt(Spin(9);\mathbb{F}_2) = 8 > 6 = wgt(Spin(9);\mathbb{F}_2).
