  A a Heintze group is a Lie group of the form $N\rtimes_\alpha \mathbb{R}$,
where $N$ is a simply connected nilpotent Lie group and $\alpha$ is a
derivation of $\mathrm{Lie}(N)$ whose eigenvalues all have positive real parts.
We show that if two purely real Heintze groups equipped with left-invariant
metrics are quasi-isometric, then up to a positive scalar multiple, their
respective derivations have the same characteristic polynomial. Using the same
thecniques, we prove that if we restrict to the class of Heintze groups for
which $N$ is the Heisenberg group, then the Jordan form of $\alpha$, up to
positive scalar multiples, is a quasi-isometry invariant.
