Faithful completely reducible representations of modular Lie algebras

The Ado-Iwasawa Theorem asserts that a finite-dimensional Lie algebra $L$ over a field $F$ has a finite-dimensional faithful module $V$. There are several extensions asserting the existence of such a module with various additional properties. In particular, Jacobson has proved that if the field has characteristic $p>0$, then there exists a completely reducible such module $V$. I strengthen Jacobson's Theorem, proving that if $L$ has dimension $n$ over the field $F$ of characteristic $p>0$, then $L$ has a faithful completely reducible module $V$ with $\dim(V) \le p^{n^2-1}$.