Derivations of the intermediate Lie algebras between the Lie algebra of diagonal matrices and that of upper triangular matrices over a commutative ring

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Let R be an arbitrary commutative ring with identity, gl(n, R) the general linear Lie algebra over R consisting of all n × n matrices over R and with the bracket operation [x, y] = xy − yx, t (resp., u) the Lie subalgebra of gl(n, R) consisting of all n × n upper triangular (resp., strictly upper triangular) matrices over R and d the Lie subalgebra of gl(n, R) consisting of all n × n diagonal matrices over R. The aim of this article is to give an explicit description of the derivation algebras of the intermediate Lie algebras between d and t.