On Riedtmann's Lie algebra of the gentle one-cycle algebra $\mathrm{\Lambda }(n-1,1,1)$

An extended version of Riedtmann’s Lie algebra of the gentle one-cycle algebra $\mathrm{\Lambda }(n-1,1,1)$ is computed and is shown to admit a Cartan decomposition by the positive roots of the root system of type ${\mathbf{B}\mathbf{C}}_n$.