Reading some notes on Lie algebras now.
Let $g$ be the Lie algebra $sl_2(\mathbb{C}))$ and let $U$ be the universal enveloping algebra.
In the course of some computation (proving the Casimir element $C = EF + FE + H^2 / 2$ to be central), the author writes:
$[E,EF] = [E,E]F + E[E,F]$.
It's not clear to me why this is true. Is it a general fact in the universal enveloping algebra of a lie algebra that $[E,FG] = [E,F]G + E[F,G]$? Or maybe something else? Or is the author using something about $sl_2(\mathbb{C})$?
I can compute: $[E,FG] = EFG - FGE$ and $[E,F]G + E[F,G] = EFG - FEG + EFG - EGF$, which doesn't do what we want.
I can also write $[E,FG] = [E, GF + [F,G]] = [E,GF] + [E,[F,G]]$, which doesn't get us anywhere.
As your computations show, it's not true in general. Nevertheless, in this particular case: $[E, E]F + E [E, F] = EEF - EEF + EEF - EFE = EEF - EFE = [E, EF]$.