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Abstract: Let be a fibre bundle with structure group G, where B is -connected and of finite dimension, . We prove that the strong L–S category of X is less than or equal to , if F has a cone decomposition of length 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 for all , which might be best possible, since we have for any prime p and . Similarly, we obtain the L–S category of for and . We remark that all the above Lie groups satisfy the Ganea conjecture on L–S category. [Copyright &y& Elsevier]

We give a cellular decomposition of the compact connected Lie group Spin(7). We also determine the L–S categories of Spin(7) and Spin(8). [Copyright &y& Elsevier]