so O(n) is the disjoint union of the subsets O(n)+ and O(n)-. 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)