  The aim of this short note is to give examples of $L^p$-$L^q$ bounded
spectral multipliers for operators involving left-invariant vector fields and
their inverses, in the settings of Engel and Cartan groups. The interest in
such examples lies in the fact that a group does not have to have flat
co-adjoint orbits, and that the considered operator is not related to the usual
sub-Laplacian. The discussed examples show how one can still obtain $L^p$-$L^q$
estimates for similar operators in such settings. As immediate consequences,
one gets the corresponding Sobolev-type inequalities and heat kernel estimates.
