Let $S^3$ denote the unit quaternions with multiplication $\mu:S^3\times S^3\rightarrow S^3$.Show that if $f_1,f_2:S^3\rightarrow S^3$ are given maps,that the composition $$S^3\xrightarrow{f_1\times f_2}S^3\times S^3\xrightarrow{\mu}S^3$$ has degree equal to $deg(f_1)+deg(f_2)$, i.e. $$deg(\mu(f_1,f_2))=deg(f_1)+deg(f_2)$$
My confusion is that I cannot calculate the degree by composion since the middle one is not a sphere and the quaternion multiplication is also a bit of complex to deal with.How to deal with that?Thanks a lot.