Waring's problem for integral quaternions

A (Lipschitz) integral quaternion is a Hamiltonian quaternion of the form a+bi+cj+dk with all of a,b,c,d∈Z. In 1946, Niven showed that the integral quaternions expressible as a sum of squares of integral quaternions are precisely those for which 2∣b,c,d; moreover, all of these are expressible as sums of three squares. Now let m be an integer with m>2, and suppose 2r∥m. We show that if r=0 (i.e., m is odd), then all integral quaternions are sums of mth powers, while if r>0, then the integral quaternions that are sums of mth powers are precisely those for which 2r∣b,c,d and 2r+1∣b+c+d. Moreover, all of these are expressible as a sum of g(m)mth powers, where g(m) is a positive integer depending only on m.