  Let $\mathfrak{g}$ be the simple Lie algebra of square matrices $(n+1)\times
(n+1)$ with zero trace. There are certain relations concerning standard
automorphisms that are considered ``folklore". One can find a complete proof of
these in Millson and Toledano Laredo's work [Transformation Groups, Vol. 10,
No. 2, 2005, pp. 217-254], but said proof is based on topological arguments
which are non-trivial. The aim of this work is to verify these relations in an
elementary way.
