Given a sequence $\overrightarrow{d}=(d_1,\dots ,d_k)$ of natural numbers, we consider the Lie subalgebra $\mathfrak{h}$ of $\mathfrak{gl}(d,\mathbb{F})$, where $d=d_1+\cdots +d_k$ and $\mathbb{F}$ is a field of characteristic 0, generated by two block upper triangular matrices D and E partitioned according to $\overrightarrow{d}$, and study the problem of computing the nilpotency degree m of the nilradical $\mathfrak{n}$ of $\mathfrak{h}$. We obtain a complete answer when D and E belong to a certain family of matrices that arises naturally when attempting to classify the indecomposable modules of certain solvable Lie algebras.

Our determination of m depends in an essential manner on the symmetry of E with respect to an outer automorphism of $\mathfrak{sl}(d)$. The proof that m depends solely on this symmetry is long and delicate.

As a direct application of our investigations on $\mathfrak{h}$ and $\mathfrak{n}$ we give a full classification of all uniserial modules of an extension of the free ℓ-step nilpotent Lie algebra on n generators when $\mathbb{F}$ is algebraically closed.

给定一个序列 $\overrightarrow{d}=(d_1,\dots ,d_{克})$ 自然数，我们考虑李子代数 $\mathfrak{H}$ 的 $\mathfrak{高尔}(d,\mathbb{F})$， 在哪里 $d=d_1+\cdots +d_{克}$ 和 $\mathbb{F}$是一个特征为 0 的场，由两个分块上三角矩阵D和E生成$\overrightarrow{d}$, 并研究计算nilradical的幂零度m的问题$\mathfrak{n}$ 的 $\mathfrak{H}$. 当D和E属于在尝试对某些可解李代数的不可分解模进行分类时自然出现的某个矩阵族时，我们会得到一个完整的答案。

我们对m 的确定在本质上取决于E相对于$\mathfrak{SL}(d)$. m仅依赖于这种对称性的证明是漫长而微妙的。

作为我们调查的直接应用 $\mathfrak{H}$ 和 $\mathfrak{n}$我们给出了n 个生成器上的自由ℓ阶幂零李代数的扩展的所有单列模块的完整分类，当$\mathbb{F}$ 是代数闭的。