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 $n\leq 7$ and give the real classification for dimension 8. 
J. M. Ancochea Bermudez, R. Campoamor-Stursberg, L. Garcia Vergnolle