Fix a Lie algebra \(L\) over a field \(\mathbb{F}\) of characteristic \(p>0\). Denote by \(S(L)\) the corresponding symmetric Poisson algebra, and let \(\mathbf{s}(L):=S(L)/I\) be the truncated Poisson algebra obtained as the quotient of \(S(L)\) by the Poisson ideal \(I=( x^p \mid x\in L) \subset S(L)\). If \(p\geq 3\), it was proven in [I. Z. Monteiro Alves and V. Petrogradsky, J. Algebra 488, 244–281 (2017; Zbl 1427.17033)] that: (1) \(S(L)\) is solvable if and only if \(L\) is abelian; (2) \(\mathbf{s}(L)\) is solvable if and only if \(L\) is solvable and its derived subalgebra \([L,L]\) is finite-dimensional. Furthermore, a conjecture was given in [loc. cit.] regarding the case of characteristic \(p=2\). The present paper corrects that conjecture by providing an explicit characterisation of \(L\) when \(S(L)\) (or \(\mathbf{s}(L)\)) is solvable and \(p=2\). More precisely, the following two results are proven (where one writes \(\langle X\rangle_{\mathbb{F}}\) for the vector space spanned by a set \(X\), and \(Z(L)\) for the centre of \(L\)).
Theorem. (Truncated case, see Theorem 1.3.) When \(p=2\), \(\mathbf{s}(L)\) is solvable if and only if \(L\) has a finite-dimensional solvable ideal \(I\) such that \(L/I=\langle x\rangle_{\mathbb{F}}\oplus A\), where \(A\) is an abelian ideal of \(L/I\) and \((\operatorname{ad}_{L/I}(x))^2=\lambda\, \operatorname{ad}_{L/I}(x)\) for some \(\lambda\in \mathbb{F}\).
Theorem. (General case, see Theorem 1.4.) When \(p=2\), \(S(L)\) is solvable if and only if \(L\) satisfies one of the following three conditions:

1.

\(L=\langle x\rangle_{\mathbb{F}}\oplus A\), where \(A\) is an abelian ideal of \(L\) and \((\operatorname{ad}_{L}(x))^2=\lambda\, \operatorname{ad}_{L}(x)\) for some \(\lambda\in \mathbb{F}\);

2.

\(L\) is nilpotent of class \(2\) (i.e. \([L,[L,L]]=0\)) and \(\dim(L/Z(L))=3\);

3.

\(L=\langle x_1,x_2,y \rangle_{\mathbb{F}}\oplus Z(L)\), with \([x_1,y]=x_2\), \([x_2,y]=x_2\) and \([x_1,x_2]\in Z(L)\).

The paper is written very precisely. The different steps for proving the above theorems heavily rely on the assumption on the characteristic \(p=2\) of the base field, as well as the following 3 results. The first one bounds the (co)dimension of \(\Delta(L):=\{x\in L \mid \dim [L,x]<\infty\}\) and its derived subalgebra in \(L\) when \(S(L)\) or \(\mathbf{s}(L)\) satisfies a multilinear Poisson identity; this is Theorem 5.6 in [loc. it.] The second one states that for fixed \(x\), the restriction of \(\operatorname{ad}_L(x)\) to an abelian ideal of \(L\) is algebraic when \(S(L)\) or \(\mathbf{s}(L)\) is solvable; this is Lemma 10.1 in [loc. it.] The third one bounds the nilpotence index of a specific Poisson ideal in a solvable Poisson algebra; this is Theorem 3.3 in [S. Siciliano and H. Usefi, J. Algebra 568, 349–361 (2021; Zbl 1472.17083)].