Suppose $\mathfrak{g}$ is a restricted Lie algebra over a field of characteristic $p>0$. Are there conditions on $\mathfrak{g}$ and its restriction which ensure that its restricted enveloping algebra is a domain? That it is a finitely generated domain? I really care about the graded case, so what if we assume further that $\mathfrak{g}$ is graded, concentrated in positive even degrees, and finite-dimensional in each degree?
Edit: for example, if the restriction is injective, does that mean that $u(\mathfrak{g})$ is a domain? What if you only know that if $x^{[p]}=0$, then $x=0$ – is that good enough? In the graded case, if the restriction is also surjective in all sufficiently large degrees, is $u(\mathfrak{g})$ a finitely generated domain?
Are there conditions on $\mathfrak{g}$ and its restriction which make $u(\mathfrak{g})$ isomorphic to an ordinary enveloping algebra $U(L)$ of some Lie algebra $L$?
