The spectrum of the Vladimirov sub-Laplacian on the compact Engel group

Let $p>3$ be a prime number. In this note, we use p-adic Gaussian integrals to calculate explicitly the unitary dual and the matrix coefficients of the Engel group over the $p$-adic integers ${\mathcal{B}}_4({\mathbb{Z}}_p)$. We use this information to calculate explicitly the spectrum of the Vladimirov sub-Laplacian, and show how it defines a globally hypoelliptic operator on ${\mathcal{B}}_4$.