Consider the Lie algebra $\mathfrak{g}$ span by $x_1,\ldots,x_n,y_1,\ldots,y_n$ with the commutator

$\left[y_i,x_i\right]=y_i, \left[x_i,x_{i+1}\right]=y_i, \left[y_i,x_{i+1}\right]=-y_i,\quad 1\leq i\leq n$

and the other brackets not mentioned are zero and the indices are mod $n$ (for instance when $n=3$ we have $\left[x_3,x_1\right]=y_3=\left[x_1,y_3\right]$).

We have that $g$ is solvable, $Z(\mathfrak{g})=0$ and $\mathfrak{h}=\langle y_i,i=1,\ldots,n\rangle$ is abelian ideal of $g$.

This Lie algebra appears in completely integrable systems, it gives a Hamiltonian formulation of the relativistic Toda lattice. I want to know if this is a very known solvable Lie algebra and what can other properties can say about it?