A note on symmetric vielbeins in bimetric, massive, perturbative and non perturbative gravities

We consider a manifold endowed with two different vielbeins E-mu(A) and L-mu(A) corresponding to two different metrics g(mu nu) and f(mu nu). Such a situation arises generically in bimetric or massive gravity (including the recently discussed version of de Rham, Gabadadze and Tolley), as well as in perturbative quantum gravity where one vielbein parametrizes the background space-time and the other the dynamical degrees of freedom. We determine the conditions under which the relation g(mu nu)E(mu)(A)L(nu)(B) = g(mu nu)E(mu)(B)L(nu)(A) can be imposed (or the Deser-van Nieuwenhuizen gauge chosen). We clarify and correct various statements which have been made about this issue. We show in particular that in D = 4 dimensions, this condition is always equivalent to the existence of a real matrix square root of g(-1) f.