  We prove that, in case $A(c)$ = the FRT construction of a braided vector
space $(V,c)$ admits a weakly Frobenius algebra $\mathfrak B$ (e.g. if the
braiding is rigid and its Nichols algebra is finite dimensional), then the Hopf
envelope of $A(c)$ is simply the localization of $A(c)$ by a single element
called the quantum determinant associated to the weakly Frobenius algebra. This
generalizes a result of the author together with Gast\'on A. Garc\'ia in
\cite{FG}, where the same statement was proved, but with extra hypotheses that
we now know were unnecessary. On the way, we describe a universal way of
constructing a universal bialgebra attached to a finite dimensional vector
space together with some algebraic structure given by a family of maps
$\{f_i:V^{\otimes n_i}\to V^{\otimes m_i}\}$. The Dubois-Violette and Launer
Hopf algebra and the co-quasi triangular property of the FRT construction play
a fundamental role on the proof.
