It is well-known that $[fX,gY]=fg[X,Y]+fX(g)Y-gY(f)X$ where $f,g$ are scalar functions and $X,Y$ are vector fields. However, it is also easy to demonstrate that the Lie bracket is anti-commutative and linear in the first argument. Why is this reasoning wrong?
$$[fX,gY] = f[X,gY] = -f[gY,X] = -fg[Y,X] = fg[X,Y]$$
I'm sure I'm making a mistake, but I can't see where. I am using only anti-commutativity and linearity in the first argument. Where is the mistake?
The bracket is $\mathbb R$-linear in both arguments, not $C^\infty$-linear.
So you can do $[aX,Y]=a[X,Y]$ for $a \in \mathbb R$, but $[fX,Y]\neq f[X,Y]$ in general for $f\colon M\to\mathbb R$.
In fact, the $\mathbb R$-linearity is an incarnation of $[fX,gY]=fg[X,Y]+fX(g)Y-gY(f)X$: $a$ gives rise to a $C^\infty$-map $\lambda_a \colon M\to \mathbb R$ via $\lambda_a(p)=a$. Then $X(\lambda_a)$ and $Y(\lambda_a)$ vanish, so you obtain $[\lambda_a X,Y] = \lambda_a[X,Y]$.