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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 (and under which conditions) 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 $$\mathrm{ind}\ \mathfrak{g}\leq \mathrm{ind}\ \mathfrak{h}+\dim\mathfrak{s},$$ and, if yes, when the above inequality is strict (i.e. has $<$ instead of $\leq$) or, on the other hand, when does it 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 above (or perhaps 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.

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 (and under which conditions) 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 $$\mathrm{ind}\ \mathfrak{g}\leq \mathrm{ind}\ \mathfrak{h}+\dim\mathfrak{s},$$ and, if yes, when the above inequality is strict (i.e. has $<$ instead of $\leq$) or, on the other hand, when does it 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 above (or perhaps 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.

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 (and under which conditions) 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 $$\mathrm{ind}\ \mathfrak{g}\leq \mathrm{ind}\ \mathfrak{h}+\dim\mathfrak{s},$$ 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.

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 (and under which conditions) 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 $$\mathrm{ind}\ \mathfrak{g}\leq \mathrm{ind}\ \mathfrak{h}+\dim\mathfrak{s},$$ and, if yes, when the above inequality is strict (i.e. has $<$ instead of $\leq$) or, on the other hand, when does it 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 above (or perhaps 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.

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.

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 (and under which conditions) 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 $$\mathrm{ind}\ \mathfrak{g}\leq \mathrm{ind}\ \mathfrak{h}+\dim\mathfrak{s},$$ 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.