Descent study of the Lie algebra of derivations of certain infinite-dimensional Lie algebras

Let g be a finite-dimensional perfect Lie algebra over a field k of characteristic 0. In infinite-dimensional Lie theory we encounter Lie algebras of the form g?(k) R, where R is a k-ring (usually a Laurent polynomial ring in finitely many variables over k), and & eacute;tale twisted forms L of g ?(k) R. Thus L is an R-Lie algebra that becomes isomorphic to the S-Lie algebra g ?(k) S after some & eacute;tale cover base ring extension S/R. The interesting infinite-dimensional Lie algebras are "built" out of L by adding a centre Z and a Lie algebra of derivations D (the affine Kac-Moody Lie algebras are the simplest examples). D, which determines Z, is a Lie subalgebra of Der(k)(L) of L. The understanding of this last Lie algebra is crucial. While the R-Lie algebra L can be given by & eacute;tale descent, the same cannot be openly said about Der(k)(L) since it is not an R-Lie algebra. In the present paper we give such a descent presentation within the more general framework of relative R/k-sheaves of Lie algebras that we believe is of independent interest.