  Starting from a finite-dimensional representation of the Yangian
$Y(\mathfrak{g})$ for a simple Lie algebra $\mathfrak{g}$ in Drinfeld's
original presentation, we construct a Hopf algebra
$X_\mathcal{I}(\mathfrak{g})$, called the extended Yangian, whose defining
relations are encoded in a ternary matrix relation built from a specific
$R$-matrix $R(u)$. We prove that there is a surjective Hopf algebra morphism
$X_\mathcal{I}(\mathfrak{g})\twoheadrightarrow Y(\mathfrak{g})$ whose kernel is
generated as an ideal by the coefficients of a central matrix $\mathcal{Z}(u)$.
When the underlying representation is irreducible, we show that this matrix
becomes a grouplike central series, thereby making available a proof of a
well-known theorem stated by Drinfeld in the 1980's. We then study in detail
the algebraic structure of the extended Yangian, and prove several
generalizations of results which are known to hold for Yangians associated to
classical Lie algebras in their $R$-matrix presentations.
