Let $U_\hbar\mathfrak{g}$ denote the Drinfeld-Jimbo quantum group associated
to a complex semisimple Lie algebra $\mathfrak{g}$. We apply a modification of
the $R$-matrix construction for quantum groups to the evaluation of the
universal $R$-matrix of $U_\hbar\mathfrak{g}$ on the tensor square of any of
its finite-dimensional representations. This produces a quantized enveloping
algebra $\mathrm{U_R}(\mathfrak{g})$ whose definition is given in terms of two
generating matrices satisfying variants of the well-known $RLL$ relations. We
prove that $\mathrm{U_R}(\mathfrak{g})$ is isomorphic to the tensor product of
the quantum double of the Borel subalgebra $U_\hbar\mathfrak{b}\subset
U_\hbar\mathfrak{g}$ and a quantized polynomial algebra encoded by the space of
$\mathfrak{g}$-invariants associated to the semiclassical limit $V$ of the
underlying finite-dimensional representation of $U_\hbar\mathfrak{g}$. Using
this description, we characterize $U_\hbar\mathfrak{g}$ and the quantum double
of $U_\hbar\mathfrak{b}$ as Hopf quotients of $\mathrm{U_R}(\mathfrak{g})$ and
as fixed-point subalgebras with respect to certain natural automorphisms. As an
additional corollary, we deduce that $\mathrm{U_R}(\mathfrak{g})$ is
quasitriangular precisely when the irreducible summands of $V$ are distinct.

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