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?

For example, some basic questions are

1. How to define the Cartan-Killing form?
2. 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($M$).)