Let $n\ge 1$ be an integer. Let $\mathfrak h^n$ be the real [Lie algebra](/page/Lie%20Algebra) with ordered basis $X_1,\dots,X_n,Y_1,\dots,Y_n,T$ whose Lie bracket is determined by
for all $1\le j,k\le n$. Let $\mathbb H^n$ be the connected simply connected Lie group with Lie algebra $\mathfrak h^n$. Define exponential coordinates by the map