Motivated by comments of the following post A question on involutions on the Lie algebra of vector fields we ask the following question:

Let $L$ be a Lie algebra. We consider the Lie subalgebra $$F_{L}=\{x\in L \mid ad(x) \text{is a finite rank operator on L}\}$$

Note that $L\to F_{L}$ is a functor on the category of Lie algebras.

1.In what reference, I can find some materials about this special Lie subalgebra and the properties of this functor(And the functor $A\to F_{A}$ in the thirth question below?

2.For a manifold $M$, what is $F_{L}$ where $L=\chi^{\infty}(M)$. Is it always the zero Lie algebra? That is:is there a manifold for which $F_{L}$ is a nontrivial Lie algebra?

3.A $C^{*}$ algebra $A$,is a Lie algebra $A$ in an obvious manner. So $F_{A}$ is a pre $C^{*}$ algebra. We denote its completion by $F_{A}$ again. Is it true to say that $F_{M_{}n}(A)\sim M_{n}(F_{A})$?

Moreover what is $F_{A}$ for for $A=B(H)$?