A generalization of Veldkamp's theorem for a class of Lie algebras

A classical theorem of Veldkamp describes the center of an enveloping algebra of a Lie algebra of a semi-simple algebraic group in characteristic $p.$ We generalize this result to a class of Lie algebras with a property that they arise as the reduction modulo $p\gg 0$ from an algebraic Lie algebra $\mathfrak{g},$ such that $\mathfrak{g}$ has no nontrivial semi-invariants in $Sym(\mathfrak{g})$ and $Sym(\mathfrak{g})^{\mathfrak{g}}$ is a polynomial algebra. As an application, we solve the derived isomorphism problem of enveloping algebras for the above class of Lie algebras.
Comment: Significantly revised, final version. To appear in Journal of Algebra