  In this article, we investigate spectral properties of the sublaplacian
$-\Delta_{G}$ on the Engel group, which is the main example of a Carnot group
of step 3. We develop a new approach to the Fourier analysis on the Engel group
in terms of a frequency set. This enables us to give fine estimates on the
convolution kernel satisfying $F(-\Delta_{G})u=u\star k_{F}$, for suitable
scalar functions $F$, and in turn to obtain proofs of classical functional
embeddings, via Fourier techniques. This analysis requires a summability
property on the spectrum of the quartic oscillator, which we obtain by means of
semiclassical techniques and which is of independent interest.
