It is necessary to perform arithmetic in F_p¹² to use an Ate pairing on a Barreto-Naehrig (BN) curve, where p is a prime given by p(z)=36z⁴+36z³+24z²+6z+1 for some integer z. In many implementations of Ate pairings, F_p¹² has been regarded as a 6th degree extension of F_p², and it has been constructed by F_p¹²=F_p²[v]/(v⁶-ξ) for an element ξ ∈ F_p² such that v⁶-ξ is irreducible in F_p²[v]. Such a ξ depends on the value of p, and we may use a mathematical software package to find ξ. In this paper it is shown that when z ≡ 7,11 (mod 12), we can universally construct F_p¹² as F_p¹²=F_p²[v]/(v⁶-u-1), where F_p²=F_p[u]/(u²+1).