Nilpotency degree of the nilradical of a solvable Lie algebra on two generators

Given a sequence $\vec d=(d_1,\dots,d_k)$ 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 $\vec d$, 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 $\ell$-step nilpotent Lie algebra on $n$ generators when $\mathbb{F}$ is algebraically closed.