Sylvester–'t Hooft generators and relations between them for $\mathfrak{sl}(n)$ and $\mathfrak{gl}(n|n)$

Among the simple finite-dimensional Lie algebras, only $\mathfrak{sl}(n)$ has two finite-order automorphisms that have no common nonzero eigenvector with the eigenvalue one. It turns out that these automorphisms are inner and form a pair of generators that allow generating all of $\mathfrak{sl}(n)$ under bracketing. It seems that Sylvester was the first to mention these generators, but he used them as generators of the associative algebra of all $n\times n$ matrices $\operatorname{Mat}(n)$. These generators appear in the description of elliptic solutions of the classical Yang–Baxter equation, the orthogonal decompositions of Lie algebras, 't Hooft's work on confinement operators in QCD, and various other instances. Here, we give an algorithm that both generates $\mathfrak{sl}(n)$ and explicitly describes a set of defining relations. For simple (up to the center) Lie superalgebras, analogues of Sylvester generators exist only for $\mathfrak{gl}(n|n)$. We also compute the relations for this case.