Diagonal Reduction Algebras of $\mathfrak{gl}$ Type

Several general questions concerning a reduction algebra, namely, rings of definition and the algorithmic efficiency of the set of ordering relations, are discussed. For reduction algebras related to the diagonal embedding of the Lie algebra $\mathfrak{gl}_n$ into $\mathfrak{gl}_n\oplus\mathfrak{gl}_n$, a stabilization phenomenon is established and a complete set of defining relations is given.