Moufang Loops of Odd Order pq3

It has been proven by F. Leong and the author (1997, J. Algebra190, 474–486) that all Moufang loops of order pαqα11 ••• qαnn are associative if p and qi are odd primes with p < q1 < ••• < qn, and• (i) α ≤ 3, α ≤ 2; or• (ii) ≥ 5, α ≤ 4, α ≤ 2.In this paper, we prove the existence of nonassociative Moufang loops of order pq3 for every pair of odd primes, p and q with q ≡ 1 (mod p).