Lusternik-Schnirelmann category of $\mathbf{Spin}{(9)}$

First we give an upper bound of $\mathrm{cat}{(E)}$, the L-S category of a principal $G$-bundle $E$ for a connected compact group $G$ with a characteristic map $\alpha: {\Sigma}V \to G$. Assume that there is a cone-decomposition $\{F_{i}\,\vert\,0 \leq i\leq m\}$ of $G$ in the sense of Ganea that is compatible with multiplication. Then we have $\mathrm{cat}{(E)} \leq\mathrm{Max}(m{+}n,m{+}2)$ for $n \geq 1$, if $\alpha$ is compressible into $F_{n} \subseteq F_{m}\simeq G$ with trivial higher Hopf invariant $H_n(\alpha)$. Second, we introduce a new computable lower bound, $\mathrm{Mwgt} {(X; {\mathbb{F}_2}})$ for $\mathrm{cat}({X})$. The two new estimates imply $\mathrm{cat}({\mathbf{Spin}{(9))}}=\mathrm{Mwgt} ({\mathbf{Spin}{(9)};{\mathbb{F}_2}}) = 8 > 6 =\mathrm{wgt}({\mathbf{Spin}{(9)};{\mathbb{F}_2}})$, where $(\mathrm{wgt}{-;R})$ is a category weight due to Rudyak and Strom.