Hadamard matrices of dimension 4, 8, and 16 can be constructed by means of the Kronecker product from the Hadamard matrix of dimension 2.
How about n = 12? The prime power q = 11 satisfies q = 3 mod 4 and so Part 3 shows that a Hadamard matrix exists for q + 1 = 12.
How about n = 20? The prime power
q = 9 satisfies q = 1 mod 4 and so Part 2 shows that a Hadamard matrix exists
for 2(q + 1) = 20.