Haile, Han and Kuo have studied certain non-commutative algebras
associated to a binary quartic or ternary cubic form.
We extend their construction to pairs of quadratic forms
in four variables, and conjecture a further generalisation to
genus one curves of arbitrary degree. These constructions give
an explicit realisation of an isomorphism relating the
Weil-Châtelet and Brauer groups of an elliptic curve