Inside of M2(C), let 1 =1 00 1,i=i00−i, j=0 1−1 0, andk=0ii0. The set of real quaternions consists of all matrices of the forma1 +bi+cj+dk, where a, b, c and d are real numbers.
(i) Prove that i2=j2=k2=−1,ij=−ji=k, jk=−kj=i and ki=−ik=j.
(ii) Show that H is a noncommutative ring with identity.
(iii) Show that H is a division ring.