  In this article we show that, under certain conditions, equality of the
Margulis-Smilga invariant spectra of two Margulis-Smilga spacetimes induce an
automorphism of the ambient affine Lie group. In particular, we show that
equality of the Margulis-Smilga invariant spectra of two Margulis-Smilga
spacetimes, coming from the adjoint representation of a real split simple
algebraic Lie group $\mathsf{G}$ with trivial center and Lie algebra
$\mathfrak{g}$, induce an automorphism of the affine group
$\mathsf{G}\ltimes\mathfrak{g}$.
