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I'm reading about $SO(n)$ in Hatcher's algebraic topology, where is said that for $n\notin\{1,2,3,4,8\}$ $SO(n)$ is a ‘twisted product’ of $SO(n − 1)$ and $S^{n−1}$.

What is a twisted product? Where can I read more about this?

Any help is welcome. Thanks in advance.

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    He means there is a nontrivial fibration $SO(n-1)\to SO(n)\to S^{n-1}$ – 2017-01-10

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