This question was raised a while ago in a blog post by Terry Tao on the Euler-Arnold equation and he called it "quite tricky".

Has anyone in the meantime tried to formulate this question precisely, or to work it out?

I'm sketching here what I suppose the precise question should be (suggestions for improvement are very welcome). Let $M$ be a smooth orientable manifold and $\mu$ a volume form. Denote by

$$\mathfrak{g}=\text{SVect(M)}=\{X\in \text{Vect}(M)|\mathcal{L}_X\mu=0\}$$

the Lie algebra of divergence-free vector fields. The adjoint action of $\mathfrak{g}$ on itself is given by

$$ad: \mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}, \quad ad(X)(Y)=[X,Y]=\mathcal{L}_XY. $$

Thus, the Cartan-Killing form evaluated on $X,Y\in\mathfrak{g}$ should be related to the product of Lie derivative operators $\mathcal{L}_X\mathcal{L}_Y$.

Some basic questions are

- How to define the Cartan-Killing form? Suppose that $M$ is compact and $g$ is a Riemannian metric on $g$. We may regard the completion of $\mathfrak{g}$ to a Hilbert space with inner product $\langle X,Y \rangle=\int_Mg(X,Y)\mu$ (if $M$ is not compact, consider the sub-algebra of square integrable vector fields). Is $\mathcal{L}_X\mathcal{L}_Y$ a trace-class operator?
- What do we know about the relationships of SVect($M$) and $M$ that would help us identify the cases where the form can potentially be described more precisely? (I‘m thinking of things of the sort that are known about the Lie algebra of all vector fields, e.g. that M is uniquely determined by the algebraic structure of Vect($M$).)