Finite presentability of the Steinberg unitary Lie algebra

In this paper, we discuss finite presentability of the Steinberg unitary Lie algebra, which is a central extension of the elementary unitary Lie algebra. For a finitely generated unital associative algebra S and n≥4, we provide the necessary and sufficient condition for the Steinberg unitary Lie algebra stun(S⊕Sop,ex,γ) to be finitely presented. When n=3, we also give the necessary and sufficient condition under certain constraints for stu3(S⊕Sop,ex,γ) to be finitely presented. Furthermore, the finite presentation of the universal covering algebra of the elementary unitary Lie algebra is investigated. Additionally, for a finitely generated commutative associative algebra S over F, where −1 has no square root, we study the finite presentability of the Lie algebra stun(S⊕iS,a―,γ) when n≥4.