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An $n$-dimensional Lie algebra $\mathfrak{g}$ over a field $\mathbb{F}$ ofcharacteristic two is said to be of Vergne type if there is a basis$e_1,\dots,e_n$ such that $[e_1,e_i]=e_{i+1}$ for all $2\leq i \leq n-1$ and$[e_i,e_j] = c_{i,j}e_{i+j}$ for some $c_{i,j} \in \mathbb{F}$ for all $i,j \ge2$ with $i+j \le n$. We define the algebra $\mathfrak{m}_0$ by its nontrivialbracket relations: $[e_1,e_i]=e_{i+1}, 2\leq i \leq n-1$, and the algebra$\mathfrak{m}_2$: $[e_1, e_i ]=e_{i+1}, 2 \le i \le n-1$, $[e_2, e_j ]=e_{j+2},3 \le j \le n-2$.  We show that, in contrast to the corresponding real and complex cases,$\mathfrak{m}_0(n)$ and $\mathfrak{m}_2(n)$ have the same Betti numbers. Wealso prove that for any Lie algebra of Vergne type of dimension at least $5$,there exists a non-isomorphic algebra of Vergne type with the same Bettinumbers.

On the Betti numbers of filiform Lie algebras over fields of characteristic two