  Let $p>3$ be a prime number. In this note, we 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 $\mathcal{B}_4$.
