Is Sl(2,C) an ideal of gl(2,C)

Question

I believe that it is but how can one show this? This is for study so any help would be great and appreciated.

Answer 1

In every Lie algebra $L$ is the commutator subalgebra $[L,L]$ an ideal. Since it is well-known that $$ [\mathfrak{gl}_n(K),\mathfrak{gl}_n(K)]=\mathfrak{sl}_n(K), $$ the claim follows.

Answer 2

Yes: $[A,B]\in\mathfrak{sl}(n, \Bbb F)$ not only when $B\in\mathfrak{sl}(n, \Bbb F)$ and $A\in\mathfrak{gl}(n, \Bbb F)$ (which would be your goal), but also for all $A,B\in\mathfrak{gl}(n, \Bbb F)$. This is due to the identity $\operatorname{tr}(AB)=\operatorname{tr}(BA)$.