  We show that a solvable real rigid Lie algebra is not completelt rigid, by
constructing an example of minimal dimension where the external torus is not
spanned by $ad$-semisimple derivations over $\mathbb{R}$. We analyze the real
forms of nilradicals of solvable rigid Lie algebras in dimensions $n\leq 7$ and
give the real classification for dimension 8.
