Equivalence of Quaternionic Heisenberg Homogeneous Quasi-norms

Let ${\mathbb{H}}_q$ denote the quaternionic Heisenberg group of dimension $(4n+3)$ with ${\mathbb{R}}^4\times {\mathbb{R}}^3$ stratification. We identify certain homogeneous norms on the group and show that any two quasi-norms on ${\mathbb{H}}_q$ are equivalent for $n<\mathrm{\infty }$.