A simple proof of the first Kac-Weisfeiler conjecture for algebraic Lie algebras in large characteristics

Given a Lie algebra $\mathfrak{g}$ of an algebraic group over a ring $S,$ we show that the first Kac-Weisfeiler conjecture holds for reductions of $\mathfrak{g} \mod p$ for large enough primes $p,$ reproving a recent result of Martin, Stewart and Topley. As a byproduct of our proof, we show that the center of the skew field of fractions of the the enveloping algebra $\mathfrak{U}\mathfrak{g}_{k}$ for a field $k$ of characteristic $p>>0$ is generated by the $p$-center and by the reduction $\mod p$ of the center of the fraction skew field of $\mathfrak{U}\mathfrak{g}.$