augmentation ideal of restricted universal enveloping algebra

Question

For restricted Lie algebra $L$ we denote its restricted universal enveloping algebra with $u(L)$. How can we prove that the augmentation ideal has codimension $1$?

Answer 1

By Jacobson's PBW-theorem we have $\dim u(L)=p^{\dim (L)}$. Now the augmentation ideal $Lu(L)=\omega(L)$ is a proper ideal of $u(L)$. Compute its dimension.