Bipartiteness characterization of finite-field nilpotent graphs

Characterize whether the nilpotent graph of a finite-dimensional Lie algebra \(L\) over \(\mathbb{F}_q\) is bipartite if and only if \(L\) is isomorphic to \(\mathfrak{t}(2,\mathbb{F}_2)\).

Background

The nilpotent graph records pairs of elements of $L\setminus nil(L)$ that generate nilpotent subalgebras. The paper gives the explicit example that the graph for $\mathfrak{t}(2,\mathbb{F}_2)$ is a disjoint union of copies of $K_2$, and is consequently bipartite.

The question asks whether this exceptional bipartite behavior characterizes the two-dimensional upper-triangular matrix Lie algebra over $\mathbb{F}_2$ among all finite-dimensional Lie algebras over finite fields.