Solvable real rigid Lie algebras are not necessarily completely solvable [Les algèbres de Lie résolubles rigides réelles ne sont pas nécessairement complètement résolubles]

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.