,\;\;\text{with}\;\; {\mathfrak{g}}_0  {=}{\mathfrak{h}}_n{\oplus}\langle{\mathpzc{R}}_{\;0}\rangle,\;\;  {\mathfrak{g}}_{{\mathpzc{p}}}=\langle{\mathpzc{R}}_{\;{\mathpzc{p}}}\rangle,$$ where ${\mathfrak{h}}_n$ is an $n$-dimensional Cartan subalgebra