The answers to this question 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 that for a Lie algebra $\mathfrak{g}$ with a Levi decomposition $\mathfrak{g}=\mathfrak{h}\triangleright\mathfrak{s}$, where $\mathfrak{s}$ is solvable and $\mathfrak{h}$ is semisimple, we have $$\mathrm{ind}\ \mathfrak{g}\geq \mathrm{ind}\ \mathfrak{h}+\dim\mathfrak{s},$$ and, if yes, when does this inequality turn into an equality?

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

Many thanks in advance.