Lie nilpotency indices of restricted universal enveloping algebras

For a restricted Lie algebra $L$ over a field of characteristic $p>0$ we study the Lie nilpotency index $t_{L}(u(L))$ of its restricted universal enveloping algebra $u(L)$. In particular, we determine an upper and a lower bound for $t_{L}(u(L))$. Finally, under the assumption that $L$ is $p$-nilpotent and finite-dimensional, we establish when the Lie nilpotency index of $u(L)$ is maximal