Schur multiplier of finite-dimensional simple Lie algebras in positive characteristic

Question

The Schur multipliers of finite simple groups are known and easily accessible: https://en.wikipedia.org/wiki/List_of_finite_simple_groups

Moreover, as a consequence of the second Whitehead's Lemma, if $L$ is a finite-dimensional simple Lie algebra over a field $\mathbb{F}$ of characteristic zero, then its Schur multiplier $H_2(L, \mathbb{F})$ is trivial. This is no longer true in positive characteristic. Now, according to the Block-Wilson-Strade-Premet classification, every simple finite-dimensional Lie algebra over an algebraically closed field of characteristic p > 3 is of classical, Cartan, or Melikian type. For my purposes, I would appreciate if I could have need any information or possible references about the following:

QUESTION Is there any explicite description of $H_2(L,\mathbb{F})$, where $L$ is a finite-dimensional simple Lie algebra over an algebraically closed field $\mathbb{F}$ of characteristic $p>3$?

I am mainly interested in the case in which $L$ is a restricted simple Lie algebra of Cartan type.

Answer 1

As $L$ is simple, it has trivial abelianization, so one has $H_1(L,\mathbb{F})=0$. Therefore, it follows from the Universal Coefficient Theorem for Lie algebras that $H_2(L, \mathbb{F})\cong H^2(L, \mathbb{F})$. For a finite-dimensional graded Lie algebra of Cartan type over an algebraically closed field of characteristic $p>3$, the second cohomology space of $L$ with coefficients in the trivial module $\mathbb{F}$ is determined in the paper

https://www.sciencedirect.com/science/article/pii/0021869392900046