Interesting subject. Yes, the map f: S^3 x S^3 -> SO(4) via f(p,q)(y) := p y q^(-1), for all y in R^4, is a homomorphic double covering of SO(4). (where SO(n) = rotation group of R^n; S^3 = group of unit quaternions.) But dim(SO(8)) = 28, so there are not enough dimensions in the domain of the corresponding map F: S^7 x S^7 -> SO(8) for it to be onto. (And it's easy to see why it's not a homomorphism.) Likewise, g: S^3 -> SO(3) via g(p)(y) := p y p^(-1) is also a homomorphic double covering, but the corresponding map G: S^7 -> SO(7) again can't be onto, since dim(SO(7)) = 21 (and again it clearly can't be a homomorphism). These things are well-known and probably explained nicely in The Octonions by Conway & Smith, QUESTION: Are there nice descriptions of the images F(S^7 x S^7) \sub SO(8), and G(S^7) \sub SO(7) ??? --Dan ---------- Warren wrote: << . . . There is a nice characterization of 2D, 3D, and 4D (and even 5D) rotations using complex #s & quaternions. But there is no such nice characterization using octonions. . . .
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