For Witt Lie Algebras over field of characterestic $p>3$ we know that $\operatorname{dim}W(n;m):=np^{|m|}$ , such that $|m|=m\_1+⋯+m\_n$ . I would like to know what is the dimension of Witt algebras over $\mathrm {GF}(2)$. Why $\operatorname{dim}W(2,1)= ?$ 



Witt Lie algebras
I think there is only a problem of notation here. The Jacobson-Witt algebra $W(m; \underline{n})$ is known to be simple of dimension $mp^{\vert n\vert}$ (where $\vert n \vert=n\_1+n\_2 \cdots + n\_m$) except when $m = 1$ and the ground field has characteristic $p=2$. In the latter case the derived subalgebra $W(1; \underline{n})^{(1)}$ is simple of dimension $2^{\vert n \vert} − 1$, provided $\vert n \vert >1$. So I think that $W(1; \underline{2})^{(1)}$ is the Lie algebra the user is really referring to. Up to isomorphisms, this is the only $3$-dimensional simple Lie algebra over $GF(2)$. It has a basis $\{a,b,c \}$ such that $[a,b]=c$, $[b,c]=a$, $[c,a]=b$. 


