The answers to [this question][1] indicate that there is a fairly vast literature on the index of Lie algebras. Unfortunately I was not able to find in this literature (or maybe to extract from it) an answer to the following simple-looking question:

>Is it true (and under which conditions) that for a Lie algebra $\mathfrak{g}$ with a [Levi decomposition][2] $\mathfrak{g}=\mathfrak{h}\triangleright\mathfrak{s}$, where $\mathfrak{s}$ is **solvable** and $\mathfrak{h}$ is **semisimple**, we have
$$\mathrm{ind}\ \mathfrak{g}\leq \mathrm{ind}\ \mathfrak{h}+\dim\mathfrak{s},$$
and, if yes, when does this inequality turn into an equality (except for the trivial case when $\mathfrak{g}$ is abelian)?  

More broadly, are there any other inequalities estimating the index of a Lie algebra from below using its Levi decomposition?

Many thanks in advance.

**NOTE:** The original question, to which Francois Ziegler provided the counterexample in his answer, concerned the opposite inequality $\mathrm{ind}\ \mathfrak{g}\geq \mathrm{ind}\ \mathfrak{h}+\dim\mathfrak{s}$, but this very answer has motivated me to modify the question.
  
  [1]: http://mathoverflow.net/questions/37101/computing-the-index-of-a-lie-algebra-what-is-known-beyond-the-reductive-case
  [2]: http://en.wikipedia.org/wiki/Levi_decomposition