Let L be a Lie algebra with a one-dimensional maximal subalgebra. Is the following true?
Over a perfect field of characteristic 0 or p > 3, every such finite-dimensional Lie algebra is either 2-dimensional, or is a form of sl(2). General structure theory seems to indicate this.



Lie algebras with a one-dimensional maximal subalgebra
If $\mathfrak{a}$ is 1-dimensional and $\mathfrak{v}$ is an irreducible $\mathfrak{a}$-module then $\mathfrak{a}$ is maximal in $\mathfrak{a}\ltimes\mathfrak{v}$. Such $\mathfrak{v}$ of dimension $\ge 3$ indeed exists if the ground field has extensions of degree $\ge 3$, and then this Lie algebra has dimension $\ge 4$ with a maximal 1-dimensional subalgebra. (I guess this are the only finite-dimensional counterexamples, at least in characteristic zero).


