  We consider a manifold endowed with two different vielbeins $E^{A}{}_{\mu}$
and $L^{A}{}_{\mu}$ 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 $g^{\mu\nu}
E^{A}{}_{\mu} L^{B}{}_{\nu} = g^{\mu\nu} E^{B}{}_{\mu} L^{A}{}_{\nu}$ can be
imposed (or the "Deser-van Nieuwenhuizen" gauge chosen). We clarify and correct
various statements which have been made about this issue.
