An Unexpected Cyclic Symmetry of $I\mathfrak{u}_n$

Abstract: We find and discuss an unexpected (to us) order $n$ cyclic group of automorphisms of the Lie algebra $I\mathfrak{u}n := \mathfrak{u}_n\ltimes\mathfrak{u}_n^\ast$, where $\mathfrak{u}_n$ is the Lie algebra of upper triangular $n\times n$ matrices. Our results also extend to $gl{n+}^\epsilon$, a ``solvable approximation'' of $gl_n$, as defined within.