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[xᵢ, xⱼ] = Σₖ₌₁ⁿ aᵢⱼᵏ xₖ satisfy the following properties:
For all i, j there exists at most one k such that aᵢⱼᵏ ≠ 0.
For all i, k there exists at most one j such that aᵢⱼᵏ ≠ 0.
This means that each bracket [xᵢ, xⱼ] yields at most one non-zero term, and each basis vector appears in at most one bracket for a fixed left argument. These bases have many interesting algebraic and geometric applications; for example, Lauret and Will proved that a basis of a nilpotent Lie algebra is stably Ricci-diagonal if and only if it is nice. However, not every Lie algebra has a nice basis, and determining whether a given Lie algebra has such a basis is not always easy.
In low dimensions, the nilpotent Lie algebras admitting a nice basis were classified by Conti and Rossi using nice diagrams. In ongoing work, we study the existence and number of non-equivalent nice bases on Lie algebras associated to graphs. These Lie algebras were introduced by Dani and Mainkar and form a well-studied class of examples, which include the free nilpotent Lie algebras.