Lusternik-Schnirelmann categories of non-simply connected compact simple Lie groups

Let $F\hookrightarrow X\to B$ be a fibre bundle with structure group $G$, where $B$ is $(d-1)$-connected and of finite dimension, $d\ge 1$. We prove that the strong L-S category of $X$ is less than or equal to $m + \frac{\dim B}{d}$,if$F$hasaconedecompositionoflength$m$ under a compatibility condition with the action of $G$ on $F$. This gives a consistent prospect to determine the L-S category of non-simply connected Lie groups. For example, we obtain $\text{cat}PU(n)\le 3(n-1)$ for all $n\ge 1$, which might be best possible, since we have $\text{cat}\mathrm{P}\mathrm{U}(p^r)=3(p^r-1)$ for any prime $p$ and $r\ge 1$. Similarly, we obtain the L-S category of $\mathrm{S}\mathrm{O}(n)$ for $n\le 9$ and $\mathrm{P}\mathrm{O}(8)$. We remark that all the above Lie groups satisfy the Ganea conjecture on L-S category.