Computing the index of a Lie algebra: what is known beyond the reductive case? Recall that an index of a Lie algebra $\mathfrak{g}$ is $\mathrm{ind}\ \mathfrak{g} := \min\limits_{\xi \in \mathfrak{g}^*} \dim \mathrm{Ann}_{\xi}$ where $\mathrm{Ann}_{\xi}=\{h\in\mathfrak{g}| \mathrm{ad}_h^*(\xi)=0\}$ is the annihilator (also known as the stabilizer) of $\xi$ with respect to the co-adjoint representation. The relevant Wikipedia article just says that if $\mathfrak{g}$ is reductive
then $\mathrm{ind}\ \mathfrak{g}=\mathrm{rank}\ \mathfrak{g}$ but I would like to build some intuition for the non-reductive case, and my googling hasn't brought about any relevant references so far. In particular, I would very much like to know:


*

*Can one say anything about the index of a solvable Lie algebra?

*What about the index of a semidirect sum (rather than the direct sum which occurs in the reductive case) $\mathfrak{g}=\mathfrak{h}\triangleright\mathfrak{a}$, where $\mathfrak{a}$ is abelian and $\mathfrak{h}$ is arbitrary? If something is known for semisimple $\mathfrak{h}$, that would be of interest too.
Many thanks in advance!
 A: The following paper of M. Rais has a formula for index of semi-direct products that you mentioned. 
M. Rais "L’indice des produits semi-directs $g\ltimes E$. 
C. R. Acad. Sci. Paris S´er. A-B 287 (1978), no. 4, A195–A197.
A: There is quite a bit of literature by now, in the classical characteristic 0 setting of finite dimensional Lie algebras.   Looking up some of the papers listed below on arXiv (usually under math.RT) and others they refer to would be a good way to get into the recent work, including some on nilpotent Lie algebras.   Beyond this, I can't answer your specific questions in detail.   But as Victor points out, study of the index is only one step.   Even in the reductive case, the rank is just one piece of information.

*

*Dmitri I. Panyushev, The index of a Lie algebra, the centraliser of a nilpotent element, and the normaliser of the centraliser, https://arxiv.org/abs/math/0107031, https://doi.org/10.1017/S0305004102006230


*A.N. Panov, On index of certain nilpotent Lie algebras, https://arxiv.org/abs/0801.3025


*Jean-Yves Charbonnel and Anne Moreau, The index of centralizers of elements of reductive Lie algebras, https://arxiv.org/abs/0904.1778, https://www.math.uni-bielefeld.de/documenta/vol-15/11.html


*Celine Righi and Rupert W. T. Yu, On the index of the quotient of a Borel subalgebra by an ad-nilpotent ideal, https://arxiv.org/abs/0908.4201, https://www.heldermann.de/JLT/JLT20/JLT201/jlt20005.htm
A: Additional references: 
V. Dergachev, Some properties of index of Lie algebras, arXiv:math/0001042
O. Yakimova, The index of centralisers of elements in classical Lie algebras, Funct. Anal. Appl. 40 (2006), N1, 42-51, arXiv:math/0407065
