On the locus of $p$-characters defining simple reduced enveloping algebras

We confirm in two cases the conjecture stating that the reduced enveloping algebra $U_\xi(\mathfrak g)$ of a restricted Lie algebra $\mathfrak g$ is simple if and only if the alternating bilinear form associated with the given $p$-character $\xi\in\mathfrak g^*$ is nondegenerate.