Isomorphism between the $R$-matrix and Drinfeld presentations of Yangian in types $B$, $C$ and $D$

It is well-known that the Gauss decomposition of the generator matrix in the $R$-matrix presentation of the Yangian in type $A$ yields generators of its Drinfeld presentation. Defining relations between these generators are known in an explicit form thus providing an isomorphism between the presentations. It has been an open problem since the pioneering work of Drinfeld to extend this result to the remaining types. We give a solution for the classical types $B$, $C$ and $D$ by constructing an explicit isomorphism between the $R$-matrix and Drinfeld presentations of the Yangian. It is based on an embedding theorem which allows us to consider the Yangian of rank $n-1$ as a subalgebra of the Yangian of rank $n$ of the same type.