Solvable real rigid Lie algebras are not necessarily completely solvable [Les alg\`ebres de Lie r\'esolubles rigides r\'eelles ne sont pas n\'ecessairement compl\`etement r\'esolubles]

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 $n\le 7$ and give the real classification for dimension 8.