Prime Lie rings of derivations of commutative rings in characteristic 2

Let R be a commutative associative ring with 1 and let Der(R) be the Lie ring of all derivations of R. Suppose that D is a Lie subring and an R-submodule of Der(R). When R is D-prime, we give necessary and sufficient conditions for D to be Lie prime. Since results of this nature are already known for rings R of characteristic different from 2, what is really new here is the characteristic 2 case.