Poisson n-Lie algebra construction from scalar matrices and commuting derivations

### What is the conjecture
Let $(\mathcal{A}, \cdot)$ be a unital commutative associative algebra over a field $\mathbb{F}$ equipped with $n+m$ pairwise commuting derivations $d_1, \dots, d_{n+m} \in \mathrm{Der}(\mathcal{A})$, where $n \ge 2$ and $m \ge 1$.

Let $A \in M_{(n+m) \times m}(\mathbb{F})$ be a fixed matrix with scalar entries $a_{r,s} \in \mathbb{F}$. Define the $n$-ary determinant bracket for $x_1, \dots, x_n \in \mathcal{A}$ by

$$
[x_1, \dots, x_n] = \det \begin{pmatrix}
d_1(x_1) & \cdots & d_1(x_n) & a_{1,1} & \cdots & a_{1,m} \\
d_2(x_1) & \cdots & d_2(x_n) & a_{2,1} & \cdots & a_{2,m} \\
\vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\
d_{n+m}(x_1) & \cdots & d_{n+m}(x_n) & a_{n+m,1} & \cdots & a_{n+m,m}
\end{pmatrix}.
$$

The source paper proves that this bracket satisfies the Leibniz rule in each argument and establishes the Filippov fundamental identity for $n = 3$ and $n = 4$ via Grassmann–Plücker relations.

**Conjecture (Cao–Normatov–Omirov, 2026):** For all integers $n \ge 2$ and $m \ge 1$, $(\mathcal{A}, \cdot, [-,\dots,-])$ forms a Poisson $n$-Lie algebra, i.e., the bracket satisfies the Filippov fundamental identity:

$$
[x_1, \dots, x_{n-1}, [y_1, \dots, y_n]] = \sum_{i=1}^n [y_1, \dots, y_{i-1}, [x_1, \dots, x_{n-1}, y_i], y_{i+1}, \dots, y_n]
$$

for all $x_1, \dots, x_{n-1}, y_1, \dots, y_n \in \mathcal{A}$.

* **Source Paper:** *Poisson $n$-Lie algebras: constructions and the structure of solvable algebras* ([arXiv:2605.01785](https://arxiv.org/abs/2605.01785), [HTML](https://arxiv.org/html/2605.01785), [PDF](https://arxiv.org/pdf/2605.01785))
* **Authors:** Xinru Cao, Zafar Normatov, Bakhrom Omirov
* **In-Paper Location:** [Conjecture 3.5 (Page 9)](https://arxiv.org/pdf/2605.01785#page=9) in Section 3 "Principal construction of Poisson $n$-Lie algebras."
* **OpenConjecture ID:** 2476 ([OpenConjecture](https://github.com/davisrbr/conjectures-arxiv))

Good to add $n = 3$ and $n=4$ (also $n=1$ and $n=2$ if those even make sense) as `research solved` and $n=5$ and the general case as `research open`

### Prerequisites needed
All required algebraic structures exist in Mathlib4:
* Derivations of an algebra: [`Mathlib.RingTheory.Derivation.Basic`](https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Derivation/Basic.html)
* Matrix determinants: [`Mathlib.LinearAlgebra.Matrix.Determinant.Basic`](https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Determinant/Basic.html)
* Commutative associative algebras: [`Mathlib.Algebra.Algebra.Basic`](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Basic.html)
* Lie algebras and generalized brackets: [`Mathlib.Algebra.Lie.Basic`](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Basic.html)

### Choose either option
- [ ] I plan on adding this conjecture to the repository
- [x] This issue is up for grabs: I would like to see this conjecture added by somebody else

This issue was generated by an AI agent and reviewed by me.