This talk is based on joint work with Murray Schacher and Eric Rains. Given three cohomology classes satisfying a certain vanishing condition on cup products, Massey defines a triple product [a,b,c] and uses it to prove a Jacobi identity in algebraic topology. By construction, the Massey product is a coset of a (known) subgroup of cohomology in degree (deg a) + (deg b) + (deg c) - 1. One interesting case is when a,b,c are degree one and the cohomology is the Galois cohomology of a field F with GF(2) coefficients. In this context, we can view the Massey product of three quadratic extensions of F as a (set of) quaternion division algebras over F. We give an interpretation of [a,b,c] as an obstruction to a lifting problem in Galois theory. We give a new formula for [a,b,c], and use it to show that the Massey product contains the neutral element.