  Let $G$ be a compact connected Lie group and $p : E\to \Sigma A$ be a
principal G-bundle with a characteristic map $\alpha : A\to G$, where $A=\Sigma
A_{0}$ for some $A_{0}$. Let $\{K_{i}{\to} F_{i-1}{\hookrightarrow} F_{i} \,|\,
1{\le} i {\le} n,\, F_{0}{=} \{\ast\} \; F_{1}{=} \Sigma{K_{1}} \; \text{and}\;
F_{n}{\simeq} G \}$ be a cone-decomposition of $G$ of length $m$ and
$F'_{1}=\Sigma{K'_{1}} \subset F_{1}$ with $K'_{1} \subset K_{1}$ which satisfy
$F_{i}F'_{1} \subset F_{i+1}$ up to homotopy for any $i$. Our main result is as
follows: we have $\operatorname{cat}(X) \le m{+}1$, if firstly the
characteristic map $\alpha$ is compressible into $F'_{1}$, secondly the
Berstein-Hilton Hopf invariant $H_{1}(\alpha)$ vanishes in $[A, \Omega
F'_1{\ast}\Omega F'_1]$ and thirdly $K_{m}$ is a sphere. We apply this to the
principal bundle $\mathrm{SO}(9)\hookrightarrow\mathrm{SO}(10)\to S^{9}$ to
determine L-S category of $\mathrm{SO}(10)$.
