
> 
> Is $\textrm{Spin}(8)$ a direct product of $\textrm{Spin}(7)$ and $S^7$?
> 
> 
> 


I met this statement in the literature, but without a reference. If it is true, where is it explicitly written?



Is $\operatorname{Spin}(8)$ a direct product of $\operatorname{Spin}(7)$ and $S^7$?
As I suspected, the statement that the bundle $\mathrm{SO}(8)\to S^7$ is a product bundle, i.e., that
$$
\mathrm{SO}(8)\simeq S^7\times\mathrm{SO}(7)\tag1
$$
as bundles over $S^7$ is in N. Steenrod's [The topology of fibre bundles](https://mathscinet.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=AUCN&pg6=TI&pg7=ALLF&pg8=ET&review_format=html&s4=Steenrod%2C%20N%2A&s5=&s6=The%20topology%20of%20fibre%20bundles&s7=&s8=All&sort=Newest&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=1&mx-pid=1688579) (as part of Theorem 8.6).


Note that Steenrod uses the notation $R\_n$ for what is more commonly notated $\mathrm{SO}(n)$, these days. (The '$R$' is for 'rotation group'. He uses $O\_n$ for the orthogonal group.) His proof, based on the octonions (which he calls 'the Cayley numbers'), is exactly what I outlined in my comment above. He does not give a reference to an earlier statement of the result, but refers to L. E. Dickson's famous *Linear Algebra* for the properties of the Cayley numbers that are used in the proof.


He does not discuss the spin groups, but, obviously, the equivalence of bundles
$$
\mathrm{Spin}(8)\simeq S^7\times\mathrm{Spin}(7)\tag2
$$
follows from (1) by passing to the respective double covers.


