Is Lie Bracket closely related to differentiation?

Question

This property of Lie bracket attracts me a lot. $$ [AB,C]=A[B,C]+[A,C]B. $$ It looks similar to the product rule of differentiation $$ (uv)'=u'v+v'u $$ The product rule can be used to define the derivative on real and complex manifolds. So I think Lie brackets can define derivatives as well. However, I struggle to find a detailed definition of derivative via Lie bracket. That $C$ is quite annoying. I need some reference to learn this definition.

And why $G'=[G,G]$ is called the derived subgroup of $G$?

Answer 1

It is a derivative of sorts: lie bracket of vector fields.

Answer 2

On the level of linear Lie algebras, the bracket is given by $$ [A,B]=AB-BA, $$ for linear operators $A,B$ in ${\rm End}(V)$. Then we have $$ [AB,C]=ABC-CAB=A(BC-CB)+(AC-CA)B=A[B,C]+[A,C]B. $$