The comaximal graph of a finite-dimensional Lie algebra

In this paper, we introduce the comaximal graph $\mathrm{\Gamma }(L)$ of a finite-dimensional Lie algebra $L$, whose vertices are the nontrivial proper Lie subalgebras of $L$ over a field $\mathbb{F}$, and two vertices $A$ and $B$ are adjacent if and only if $⟨A,B⟩=L$. We establish general structural properties, including a characterization of isolated vertices via the Frattini subalgebra and a criterion for completeness in terms of $\mu$-algebras. We classify $\mathrm{\Gamma }(L)$ for all Lie algebras of dimension at most three over a finite field ${\mathbb{F}}_q$, providing an explicit description in each case. The resulting graphs exhibit a rich range of behaviors, depending on the structure of the derived algebra and the action of $\mathrm{ad}x$. For $L\cong {\mathfrak{s}\mathfrak{l}}_2({\mathbb{F}}_q)$, we determine several graph invariants, including the degree sequence, clique number, chromatic number, domination number, diameter, and radius, and show that $\mathrm{\Gamma }(L)$ is connected and non-planar. The graph contains a large clique formed by the nonsplit semisimple lines together with the Borel subalgebras, while the nilpotent and split semisimple lines have a more restricted adjacency structure governed by their containment in Borel subalgebras.