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I did not check the details but the matrix you get may be correct. Note that a from F.<a> = GF(2^4) is a zero of some degree-4 polynomial over GF(2) and thus powers of a are reduced modulo that polynomial (and their reduced form depends on that polynomial). For example, on SageCell the following code

F.<a> = GF(2^4)
print( F.modulus() )
for d in (1..12):
    print(f'a^{d} = {a^d}')

prints

x^4 + x + 1
a^1 = a
a^2 = a^2
a^3 = a^3
a^4 = a + 1
a^5 = a^2 + a
a^6 = a^3 + a^2
a^7 = a^3 + a + 1
a^8 = a^2 + 1
a^9 = a^3 + a
a^10 = a^2 + a + 1
a^11 = a^3 + a^2 + a
a^12 = a^3 + a^2 + a + 1