Bipartiteness classification of finite-field nilpotent graphs

Classify the finite-dimensional Lie algebras L over finite fields $F_q$ whose nilpotent graph is bipartite, and determine whether bipartiteness occurs if and only if $L$ is isomorphic to $mathfrak{t}(2,\mathbb{F}_2)$.

Background

The nilpotent graph has vertices given by the elements outside nil(L), with adjacency defined by generation of a nilpotent Lie subalgebra. The paper computes the graph for the upper-triangular Lie algebra $\mathfrak{t}(2,\mathbb{F}_2)$ and observes that it is a disjoint union of three copies of $K_2$, hence bipartite. It leaves open whether this example is the unique finite-dimensional Lie algebra over a finite field whose nilpotent graph is bipartite.