Lie algebra $ \mathfrak{g} $ for a Lie group $ \mathcal{G}$ is closed under commutation. Also, the elements of Lie Algebra form a Linear Vector Space(LVS).
Firstly, when is it allowed to define an Inner product (IP) in this vector space ? What is the characteristic of a Group for which I can't define and IP for the corresponding Lie algebra ? (I have heard about this Killing form, but not understood it completely).
Secondly, if I now define a LVS with IP, then there has to be a metric for this space (so is it a metric space ? it seems like yes !!). If there is a metric, I can again define a set of transformations that will leave the metric (or line element) of the space invariant, am I right ? (Again, I think I am referring to Casimir elements of the group, but I am not sure).
PS :I am sorry if these questions are silly. I am a physics student, and hence not able to get a concrete mathematical structure of these things. I would be quite happy if the answer is a little physics(sy).
Firstly $\kappa (x,y)=tr (ad (x)ad(y))$ is the Killing form, where $ad(x)y=[x,y]$. It is a symmetric bilinear form. In characteristic zero it is non-degenerate if and only if the Lie algebra is semisimple. For nilpotent Lie algebras, for example, the Killing form is just zero. But speaking of physics, the Heisenberg Lie algebra is nilpotent.
Secondly, semisimple Lie algebras with this inner product become a metric space, yes. There a re several good books with the title "Lie Groups and Lie Algebras for Physicists", which give you much more background.