So o(n) is the disjoint union of the subsets o(n)+ and o( the important subgroup so(n) o(n)+ o(n) is the n x n special orthogonal group. one of the main reasons for the study of the orthogonal groups o(n) and so(n) is their relationship with isometries, where an isometry of rt is a distance-preserving bijection f: rnr", i. e., lf(x) - f(y)l x-yl (x, y e r"). 4.7. the group of unit quaternions has an r-linear action on h given by a) by identifying h with r4 using the basis [i, j, k,1), show that this defines a lie homomorphism sp(1)so(4) b) show that this action restricts to an action of sp(1) on the space of pure quaternions and by identifying this with r3 using the ba- sis fi, j, k, show that this defines a surjective lie homomorphisrm a: sp(1)- s0(3) . show that ker ? (1-1)